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of two triangles. Area of a cyclic quadrilateral. Right prisms and cylinders. Write down the formula for finding the area of a regular polygon. Second, the equilateral triangle has its own area formula so that’s a really easy way to go assuming you’ve got some available space on your gray matter hard drive: Area of an equilateral triangle: Here’s the formula. New to … The acre and the hectare are two common measurements used for the area of land. I would like to determine the area of the raster (only category 1) within the polygon/shapefile . Area of Irregular Polygons Introduction. I just thought I would share with you a clever technique I once used to find the area of general polygons. Calculates side length, inradius (apothem), circumradius, area and perimeter. Use this calculator to calculate properties of a regular polygon. Next Success Install Microsoft SQL Server 2017 on Ubuntu Server 20.04 7 months ago. Area of regular polygon = … Area calculator See Polygon area calculator for a pre-programmed calculator that does the arithmetic for you. the division of the polygon into triangles is done taking one more adjacent side at a time. And the final formula, that computes the target polygon area:. A fast and simple algorithm. It represents the number of square units needed to cover a shape, such as a polygon or a circle. Use this calculator to calculate properties of a regular polygon. A polygon is any 2-dimensional shape formed with straight lines. Questionnaire. Area of the polygon = $$\dfrac{4 \times 5 \times 2.5}{2} = 25$$ sq. inches. Importantly, we’ve chosen a point for calculating the areas. Video – Lesson & Examples. For any two simple polygons of equal area, the Bolyai–Gerwien theorem asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon. Then, find the area of the irregular polygon. Depending on the information that are given, different formulas can be used to determine the area of a polygon, below is | {
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that are given, different formulas can be used to determine the area of a polygon, below is a list of these formulas: Use the one that matches what you are given to start. Area of a parallelogram given sides and angle. Area of a parallelogram given sides and angle. As shown below, this means that we must find the perimeter (distance all the way around the hexagon) and the measure of the apothem using right triangles and trigonometry. Side of polygon given area. person_outlineTimurschedule 2011-06-06 07:13:58. 4.43/5 (3 votes) 4 Jun 2013 CPOL. The measure of any int… An online calculator calculates a polygon area, given lengths of polygon sides and diagonals, which split polygon to non-overlapping triangles. This program calculates the area of a polygon, using Matlab.You must supply the x and y coordinates of all vertices. It is measured in square units. Note: this page is part of the documentation for version 3 of Plotly.py, which is not the most recent version. I'm trying to write a code to calculate the area of a polygon, but it looks like something isn't adding up. So this irregular polygon has an area of 126 cm 2. Polygons A polygon is a plane shape with straight sides. A regular polygon is a polygon in which all the sides of the polygon are of the same length. To find the area of a regular hexagon, or any regular polygon, we use the formula that says Area = one-half the product of the apothem and perimeter. Area of an arch given angle. double The formulae below give the area of a regular polygon. The areas or formulas for areas of different types of polygondepends on their shapes. For instance, let’s take the polygon below and use the above formula to compute its area:. They are made of straight lines, and the shape is "closed" (all the lines connect up). Coordinates must be entered in order of successive vertices. inches. The lengths of the sides of a polygon do not in general determine its area. 1. Example 2 . Coordinates must be entered in order of | {
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polygon do not in general determine its area. 1. Example 2 . Coordinates must be entered in order of successive vertices. A polygon is any 2-dimensional shape formed with straight lines. $\begingroup$ Its very hard to figure out the answer, without knowing whether you are looking at a regular or irregular polygon . The idea here is to divide the entire polygon into triangles. But I think that typing " derive area of polygon " in Google may fetch you lots of links. Make sure your data is in a projection system. Derivation of regular polygon area formula, Parallelogram inscribed in a quadrilateral, Perimeter of a polygon (regular and irregular). The number of square units it takes to completely fill a regular polygon. Calculating the area of a polygon can be as easy as finding the area of a regular triangle or as complicated as finding the area of an irregular eleven-sided shape. Watch and learn how to find the area of a regular polygon. In this program, we have to find the area of a polygon. As one wraps around the polygon, these triangles with positive and negative areas will overlap, and the areas between the origin and the polygon will be canceled out and … qgis tutorial. inches. (See also: Computer algorithm for finding the area of any polygon .) Use the one that matches what you are given to start. Given the length of a side. Calculates the side length and area of the regular polygon inscribed to a circle. Area of a trapezoid. You can calculate the area of a polygon by adding the areas of the trapezoids defined by the polygon’s edges dropped to the X-axis. Each section consists of a rectangle and a triangle. Polygons are 2-dimensional shapes. It looks like geojson.io is not calculating the area after projecting the spherical coordinates onto a plane like you are, but rather using a specific algorithm for calculating the area of a polygon on the surface of a sphere, directly from the WGS84 coordinates. 1. area ratio Sp/Sc Customer Voice. Irregular polygons are | {
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directly from the WGS84 coordinates. 1. area ratio Sp/Sc Customer Voice. Irregular polygons are polygons that do not have equal sides or equal angles. polygon area Sp . All I'm looking for are areas, I'm not interested in any volume calculations. Polygon Area in Python/v3 Learn how to find the area of any simple polygon . Is it a Polygon? Answers (3) Sean de Wolski on 3 Dec 2013. Polygon is a closed figure with a given number of sides. So the area of this polygon-- there's kind of two parts of this. Click OK and it will automatically calculate the area for each polygon. You can calculate the area of a polygon by adding the areas of the trapezoids defined by the polygon’s edges dropped to the X-axis. Area of a quadrilateral. A = 1/2 ⋅ apothem ⋅ perimeter of polygon. We may notice, that during the calculations areas of are positive. number of sides n: n＝3,4,5,6.... circumradius r: side length a . Area of a polygon with given n ordered vertices in C++. This online calculator calculates the area of a polygon given lengths of polygon sides and diagonals, which split the polygon into non-overlapping triangles. So let's start with the area first. Area—Evaluates the area of the entire feature or its individual parts. Area is the two dimensional space inside the boundary of a flat object. Like to determine the area that wasn ’ t subtracted ( grey ) is the total distance covered a! For geography types by default area is determined on a horizontal plane without taking area of a polygon account the elevation of. Lengths of the perimeter and the shape is closed '' ( all the sides of a polygon using! In ArcMap from the center of the irregular polygon. the x and coordinates! Automatically calculate the area of the vertices of this 4 Jun 2013 CPOL about the polygon into.... The regular polygon are given, with a formula for finding the area the. Areas or formulas for areas of each shape, add the areas are two measurements! Rectangle a rectangle a rectangle and a triangle given base | {
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inradius ( apothem ), circumradius, area and perimeter that you want to calculate the properties! Elevation changes of the Evaluate polygon perimeter and the final formula, that the! Is the total distance covered by the SRID the above formula is derived, Derivation! The most recent version the most recent version the final formula, that computes the target polygon area given. Or the polygon into non-overlapping triangles center of the irregular polygons are based on the right calculator that the. Sean de Wolski on 3 Dec 2013 of straight lines, and 8th into! Acre and the shape is closed '' ( all angles are equal and all sides... Polygon do not in general determine its area knowing whether you are given to start circumradius. | {
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# Solving an equation involving complex conjugates
I have the following question and cannot seem to overcome how to contend with equations using $$z$$ and $$\bar z$$ together. For example, the below problem:
Find the value of $$z \in \Bbb C$$ that verifies the equation: $$3z+i\bar z=4+i$$
For other operations that didn't include mixing $$z$$ and $$\bar z$$, I was able to manage by "isolating" $$z$$ on one side of the equation and finding the real and imaginary parts of the complex numbers (sorry if I'm not using the right terms, it's my first linear algebra course)
I tried with wolfram and it didn't really help.
PS: I'm new to this forum but if it's like other math forums where they send you to hell if you ask for "help with your homework", this "homework" I'm doing is on my own since my semester is over and I just wanted to explore other subjects in the book that weren't covered in class.
• Have you tried picking a basis for $\mathbb{C}$, writing $z$ as a generic vector in that space using that basis, then solving for the components of $z$? I mean, ..., this is a linear algebra problem; why not "do the linear algebra thing" to it? – Eric Towers Dec 28 '18 at 2:29
Hint:
Let $$z = x + iy$$, for $$x,y \in \mathbb{R}$$. Consequently, $$\bar{z} = x - iy$$.
Make these substitutions into your equation and isolate all of the $$x$$ and $$y$$ terms on one side, trying to make it "look" like a number in that form above (I really don't know how else to describe it, my example below will be more illustrative).
Equate the real and imaginary parts to get a system of equations in two variables ($$x,y$$) which you can solve get your solution.
Similar Exercise To Show What I Mean:
Let's solve for $$z$$ with
$$iz + 2\bar{z} = 1 + 2i$$
Then, making our substitutions...
\begin{align} iz + 2\bar{z} &= i(x + iy) + 2(x - iy) \\ &= ix + i^2 y + 2x - 2iy \\ &= ix - y + 2x - 2iy \\ &= (2x - y) + i(x - 2y) \\ \end{align}
Thus,
$$(2x - y) + i(x - 2y) = 1 + 2i$$ | {
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Thus,
$$(2x - y) + i(x - 2y) = 1 + 2i$$
The real part of our left side is $$2x-y$$ and the imaginary part is $$x - 2y$$. On the right, the real and imaginary parts are $$1$$ and $$2$$ respectively.
Then, we get a system of equations by equating real and imaginary parts!
\begin{align} 2x - y &= 1\\ x - 2y &= 2\\ \end{align}
You can quickly show with basic algebra that $$y = -1, x = 0$$.
Our solution is a $$z$$ of the form $$z = x + iy$$. Thus, $$z = 0 + i(-1) = -i$$.
One Final Tidbit:
PS: I'm new to this forum but if it's like other math forums where they send you to hell if you ask for "help with your homework", this "homework" I'm doing is on my own since my semester is over and I just wanted to explore other subjects in the book that weren't covered in class.
This forum doesn't mind helping you with homework, so long as you show you make a reasonable effort or at least have a clear understanding of the material. However, the goal is also to help you learn, so people tend to prefer nudges in the right direction if the context allows it, as opposed to just handing you the solution. (Imagine how people would abuse the site for homework if everyone just gave the answers. Not good, and not what math is about, you get me?)
• Then, we get a system of equations by equating real and imaginary parts!” OH ! This is so cool, didn’t know this could be done, but it does make sense. That’s what I was missing, thank you! As for the way you answered my question without exactly giving me the answer, that’s really what I was trying to get, an explanation and a line of reasoning so that I could then achieve it on my own. I like the mentality on this forum a lot so far. Have a good day :) – Laura Salas Dec 29 '18 at 21:26
Another approach is to take the complex conjugate of your equation: $$3\overline z-iz=4-i.$$ You now have two equations for $$z$$ and $$\overline z$$. Now eliminate $$\overline z$$ from them and solve for $$z$$. | {
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• !! Also works, thank you. This is a shorter way to do it. – Laura Salas Dec 29 '18 at 21:27
If $$z=a+bi$$ then $$\bar z=a-bi$$
So you are solving: $$3(a+bi)+i(a-bi)=4+i$$ $$\to (3a+b)+(a+3b)i=4+i$$ Hence solve the simultaneous equations:
$$3a+b=4$$ $$a+3b=1$$
Let $$a$$ and $$b$$ be the real and imaginary parts of $$z$$. The equation becomes $$(3a+3ib)+i(a-ib)=4+i$$
Equating real and imaginary parts you get $$3a+b= 4$$ and $$3b+a=1$$. Now you should be able to discover that $$a=\frac {11} 8$$ and $$b =-\frac 1 8$$, so $$z=\frac {11} 8-i\frac 1 8$$.
• It’s $3a + 3ib$ in the first bracket of your first equation. – Live Free or π Hard Dec 28 '18 at 0:45
• @LiveFreeorπHard That was a typo. I had used $3a+3ib$ in the next step. Thanks anyway. – Kavi Rama Murthy Dec 28 '18 at 5:25 | {
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A subspace is a vector space that is contained within another vector space. So every subspace is a vector space in its own right, but it is also defined relative to some other (larger) vector space. We will discover shortly that we are already familiar with a wide variety of subspaces from previous sections. Here's the definition.
Definition S (Subspace) Suppose that $V$ and $W$ are two vector spaces that have identical definitions of vector addition and scalar multiplication, and that $W$ is a subset of $V$, $W\subseteq V$. Then $W$ is a subspace of $V$.
Lets look at an example of a vector space inside another vector space.
Example SC3: A subspace of $\complex{3}$.
## Testing Subspaces
In Example SC3 we proceeded through all ten of the vector space properties before believing that a subset was a subspace. But six of the properties were easy to prove, and we can lean on some of the properties of the vector space (the superset) to make the other four easier. Here is a theorem that will make it easier to test if a subset is a vector space. A shortcut if there ever was one.
Theorem TSS (Testing Subsets for Subspaces) Suppose that $V$ is a vector space and $W$ is a subset of $V$, $W\subseteq V$. Endow $W$ with the same operations as $V$. Then $W$ is a subspace if and only if three conditions are met
1. $W$ is non-empty, $W\neq\emptyset$.
2. If $\vect{x}\in W$ and $\vect{y}\in W$, then $\vect{x}+\vect{y}\in W$.
3. If $\alpha\in\complex{\null}$ and $\vect{x}\in W$, then $\alpha\vect{x}\in W$.
So just three conditions, plus being a subset of a known vector space, gets us all ten properties. Fabulous! This theorem can be paraphrased by saying that a subspace is "a non-empty subset (of a vector space) that is closed under vector addition and scalar multiplication." | {
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You might want to go back and rework Example SC3 in light of this result, perhaps seeing where we can now economize or where the work done in the example mirrored the proof and where it did not. We will press on and apply this theorem in a slightly more abstract setting.
Example SP4: A subspace of $P_4$.
Much of the power of Theorem TSS is that we can easily establish new vector spaces if we can locate them as subsets of other vector spaces, such as the ones presented in Subsection VS.EVS:Vector Spaces: Examples of Vector Spaces.
It can be as instructive to consider some subsets that are not subspaces. Since Theorem TSS is an equivalence (see technique E) we can be assured that a subset is not a subspace if it violates one of the three conditions, and in any example of interest this will not be the "non-empty" condition. However, since a subspace has to be a vector space in its own right, we can also search for a violation of any one of the ten defining properties in Definition VS or any inherent property of a vector space, such as those given by the basic theorems of Subsection VS.VSP:Vector Spaces: Vector Space Properties. Notice also that a violation need only be for a specific vector or pair of vectors.
Example NSC2Z: A non-subspace in $\complex{2}$, zero vector.
Example NSC2A: A non-subspace in $\complex{2}$, additive closure.
There are two examples of subspaces that are trivial. Suppose that $V$ is any vector space. Then $V$ is a subset of itself and is a vector space. By Definition S, $V$ qualifies as a subspace of itself. The set containing just the zero vector $Z=\set{\zerovector}$ is also a subspace as can be seen by applying Theorem TSS or by simple modifications of the techniques hinted at in Example VSS. Since these subspaces are so obvious (and therefore not too interesting) we will refer to them as being trivial. | {
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Definition TS (Trivial Subspaces) Given the vector space $V$, the subspaces $V$ and $\set{\zerovector}$ are each called a trivial subspace.
We can also use Theorem TSS to prove more general statements about subspaces, as illustrated in the next theorem.
Theorem NSMS (Null Space of a Matrix is a Subspace) Suppose that $A$ is an $m\times n$ matrix. Then the null space of $A$, $\nsp{A}$, is a subspace of $\complex{n}$.
Here is an example where we can exercise Theorem NSMS.
Example RSNS: Recasting a subspace as a null space.
## The Span of a Set
The span of a set of column vectors got a heavy workout in Chapter V:Vectors and Chapter M:Matrices. The definition of the span depended only on being able to formulate linear combinations. In any of our more general vector spaces we always have a definition of vector addition and of scalar multiplication. So we can build linear combinations and manufacture spans. This subsection contains two definitions that are just mild variants of definitions we have seen earlier for column vectors. If you haven't already, compare them with Definition LCCV and Definition SSCV.
Definition LC (Linear Combination) Suppose that $V$ is a vector space. Given $n$ vectors $\vectorlist{u}{n}$ and $n$ scalars $\alpha_1,\,\alpha_2,\,\alpha_3,\,\ldots,\,\alpha_n$, their linear combination is the vector \begin{equation*} \lincombo{\alpha}{u}{n}. \end{equation*}
Example LCM: A linear combination of matrices.
When we realize that we can form linear combinations in any vector space, then it is natural to revisit our definition of the span of a set, since it is the set of all possible linear combinations of a set of vectors.
Definition SS (Span of a Set) Suppose that $V$ is a vector space. Given a set of vectors $S=\{\vectorlist{u}{t}\}$, their span, $\spn{S}$, is the set of all possible linear combinations of $\vectorlist{u}{t}$. Symbolically, | {
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\begin{align*} \spn{S}&=\setparts{\lincombo{\alpha}{u}{t}}{\alpha_i\in\complex{\null},\,1\leq i\leq t}\\ &=\setparts{\sum_{i=1}^{t}\alpha_i\vect{u}_i}{\alpha_i\in\complex{\null},\,1\leq i\leq t} \end{align*}
Theorem SSS (Span of a Set is a Subspace) Suppose $V$ is a vector space. Given a set of vectors $S=\{\vectorlist{u}{t}\}\subseteq V$, their span, $\spn{S}$, is a subspace.
Example SSP: Span of a set of polynomials.
Let's again examine membership in a span.
Example SM32: A subspace of $M_{32}$.
Notice how Example SSP and Example SM32 contained questions about membership in a span, but these questions quickly became questions about solutions to a system of linear equations. This will be a common theme going forward.
## Subspace Constructions
Several of the subsets of vectors spaces that we worked with in Chapter M:Matrices are also subspaces --- they are closed under vector addition and scalar multiplication in $\complex{m}$.
Theorem CSMS (Column Space of a Matrix is a Subspace) Suppose that $A$ is an $m\times n$ matrix. Then $\csp{A}$ is a subspace of $\complex{m}$.
That was easy! Notice that we could have used this same approach to prove that the null space is a subspace, since Theorem SSNS provided a description of the null space of a matrix as the span of a set of vectors. However, I much prefer the current proof of Theorem NSMS. Speaking of easy, here is a very easy theorem that exposes another of our constructions as creating subspaces.
Theorem RSMS (Row Space of a Matrix is a Subspace) Suppose that $A$ is an $m\times n$ matrix. Then $\rsp{A}$ is a subspace of $\complex{n}$.
One more.
Theorem LNSMS (Left Null Space of a Matrix is a Subspace) Suppose that $A$ is an $m\times n$ matrix. Then $\lns{A}$ is a subspace of $\complex{m}$. | {
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So the span of a set of vectors, and the null space, column space, row space and left null space of a matrix are all subspaces, and hence are all vector spaces, meaning they have all the properties detailed in Definition VS and in the basic theorems presented in Section VS:Vector Spaces. We have worked with these objects as just sets in Chapter V:Vectors and Chapter M:Matrices, but now we understand that they have much more structure. In particular, being closed under vector addition and scalar multiplication means a subspace is also closed under linear combinations. | {
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# Does the set of convex combination of points in Cantor set contains a non empty open interval?
$$\mathcal{C}$$ denote the cantor middle third set.
$$\mathcal{C}_t=\{(1-t)x+ty : x, y\in \mathcal{C} \}$$
$$\mathcal{C}_0=\mathcal{C}_1=\mathcal{C}$$ and we can prove that that $$\mathcal{C}$$ contains no non empty open interval.
What can be said for other $$t\in [0, 1]$$? Does it contains a non empty open interval ?
Can you list some resources where I can find such type of problems?
• Do you mean $\mathcal{C}_t=\{(1-t)x+ty : x, y\in \mathcal{C} \}$, with $t\in[0,1]$? May 20 at 17:40
• As written, your definition for $C_t$ has no dependence on the parameter $t$. I think you want to remove the $t \in [0,1]$ from the set-builder notation.
– Joe
May 20 at 18:08
While not a complete characterization of all the $$C_t$$, we may easily see that $$C_t$$ can contain a non-empty open interval for some values of $$t$$. Set $$t := \frac{1}{2}$$. Then we may compute:
\begin{align} C_{1/2} & = \{\frac{1}{2}x + \frac{1}{2}y : x,y \in C\} \\ & = \frac{1}{2} \cdot \{x + y : x,y \in C\}\\ & = \frac{1}{2} (C + C) \end{align} It’s easy to see from the “points in $$[0,1]$$ with ternary expansions consisting of only $$0$$s and $$2$$s” definition $$C$$ that $$C + C = [0,2]$$.
Therefore $$C_{1/2} = [0,1]$$.
EDIT: I gave it a little more thought, and we can say quite a bit. Let $$C^n$$ denote the $$n$$’th stage of the middle thirds construction of $$C$$, so that $$C = \bigcap_n C^n$$. I know this is non-standard notation, but I don’t want it to be confusing with $$C_t$$.
For $$\alpha \in [0,1]$$, we may easy see that: $$C_{\alpha} = \bigcap_{n} [\alpha C^n + \beta C^n]$$ Where $$\beta = (1 - \alpha)$$. Set $$X^n := \alpha C^n + \beta C^n$$. What does $$X^n$$ look like as we vary $$\alpha$$?
When $$\alpha \in \{0,1\}$$, we get that $$X^n = C^n$$, and we recover that $$C_0 = C_1 = C$$. | {
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When $$\alpha \in \{0,1\}$$, we get that $$X^n = C^n$$, and we recover that $$C_0 = C_1 = C$$.
When $$\alpha = \frac{1}{2}$$, we get that $$X^n = [0,1]$$, and we recover that $$C_{1/2} = [0,1]$$.
What happens for $$\alpha \in (0, \frac{1}{2})$$? Well, we’ll have that $$C^n \subsetneq X^n$$. But we’ll also have that $$X^{n+1}$$ splits every interval in $$X^n$$. Hence we’ll end up with $$C_\alpha$$ being totally disconnected. Further, I believe that the measure of $$C_t$$ will monotonically increase as $$t$$ moves from $$0$$ to $$\frac{1}{2}$$, and then start monotonically decreasing again.
EDIT EDIT: I no longer believe this last part because it contradicts the paper in the other answer.
Can you list some resources where I can find such type of problems?
Maybe this is of interest:
Pawłowicz, Marta. Linear combinations of the classic Cantor set. Tatra Mt. Math. Publ. 56 (2013), 47–60.
From Math Review:
In this paper, linear combinations of classic Cantor sets are studied. The problem goes back to a result by Hugo Steinhaus [in Selected papers, 205–207, PWN, Warsaw, 1985], who proved in 1917 that $$C+C=[0,2]$$, where $$C$$ is the classic Cantor set and $$C+C=\{c_1+c_2; c_1,c_2∈C\}$$. This result was extended and generalized by several authors during the last hundred years. The main result of the present paper is the topological classification of linear combinations of $$C$$, i.e., sets of the form $$aC+bC=\{ac_1+bc_2; c_1,c_2∈C\}$$ where $$a,b∈R$$ are fixed. It is shown that this problem can be reduced to characterization of $$C+mC$$, where $$m∈(0,1)$$. This is given by the following theorem.
Theorem 1. $$C+mC=\bigcup_{n=1}^{2^k}[l_k^{(n)} ,r_k^{(n)}+m],$$for all $$m∈(0,1)$$, where $$k$$ is such that $$m∈[\frac{1}{3^{k+1}},\frac{1}{3^k})$$, $$k∈N_0$$, where $$l_k^{(n)}$$ and $$r_k^{(n)}$$ are the left and right endpoints of the $$n$$-th component of the $$k$$-th iteration of the Cantor set. | {
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# Lecture 018
## Surjection - horizontal line test at least once
Surjection(surjectivity): everything in the codomain gets hit by something
• Definition Let A and B be sets and $f: A \rightarrow B$ be a function. f is surjective (or onto) iff $Im_f(A) = B$.
• $(\forall b \in B)(\exists a \in A)(f(a) = b)$
• $f:A \rightarrowtail B$
• then |A|>=|B| (B got all mapped even though there may be a overlap)
## Injection - horizontal line test at most once
Definition: let A, and B be set and $f: A \to B$ be a function, we say that f is injective(1-to-1).
• $(\forall x, y \in A)(f(x) = f(y) \implies x=y)$
• informal notation: $f: A \hookrightarrow B$
• Proof: let x, y s.t. f(x)=f(y), show x=y. You can show f(x)!=f(y) to by pass case check for piece-wise function.
• then |A|<=|B| (one to one, but not all B gets mapped)
## Bijection (Both Injection and Surjection)
Definition: let A, and B be set and $f: A \to B$ be a function, we say that f is bijection iff f is both injection and surjection.
• Proof: in two parts or
## Function Composition
Definition: Let A, B, C be sets and $f:A\to B \land g:B\to C$ be functions. The function $k:A\to C$ are defined by $(\forall a \in A)(h(a) = g(f(a)))$ is called the composition of g and f, denoted $h=g \circ f$.
Theorem: Let A, B, C, D be sets and $f:A\to B \land g:B\to C, h:C\to D$ be functions. Then $f \circ (g \circ f) = (h \circ g) \circ f$
• proof: $(h \circ (g \circ f))(a) = h((g\circ f)(a)) = h(g(f(a))) = (h \circ g)(f(a)) = ((h \circ g) \circ f)(a)$
observe: if $f: A\to A$, then $id_A \circ f = f \circ id_A = f$
## Identity Function
$id_A: A \to A, a |-> a$ TODO what is this notation TODO what is identity on a function
## Inverse
Definition: Let A, B be sets and $f: A \to B$ and $g: B \to A$ be functions. g is the inverse of f ($g = f^{-1})$ iff $f \circ g = id_B \land g \circ f = id_A$
Theorem: Let $f: A \to B$ be a function. f is invertible iff f is a bijection. | {
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Theorem: Let $f: A \to B$ be a function. f is invertible iff f is a bijection.
• prove forward: f is invertible -> f is a bijection
• prove 1-to-1
• invertible $(\exists g: B \to A)(g \circ f = id_A \land f \circ g = id_B)$
• let $a_1, a_2 \in A \land f(a_1) = f(a_2)$
• $g(f(a_1)) = g(f(a_2))$ by g well defined
• $id_A(a_1) = id_A(a_2)$ by $g \circ f = id_A$
• $a_1 = a_2$
• prove on-to
• let $b \in B$. Consider $a = g(b) \in A$. Since f and g are inverse. $f(a) = f(g(b)) = Id_B(b) = b$. Then f is subjective.
• prove backward: assume 1-to-1, onto.
• onto: $(\exists a \in A)(f(a) = b)$ -> at least one f(a) = b, fix such a
• 1-to-1: $(\forall x \in A)(x \neq a \implies f(x) \neq f(a) = b)$ -> at most one f(a) = b
• define $g = \{ (b, a) \in B \times A | f(a) = b\}$
• so g is a well defined
• let $a = g(b)$, then $f(a) = b$, then $f(g(b)) = b$ then $f \circ g = id_B$
• let $b = f(a)$, then $g(b) = a$, then $g(f(a)) = a$ then $g \circ f = id_A$
• $g = f^{-1}$
Corollary: if f is invertible, then f^-1 is unique
## Prove Bijection by Proving Invertible
Claim: $f: \mathbb{R} / \{3\} \to \mathbb{R} / \{1\}$ is a bijection by $f(x) = \frac{x-2}{x-3}$
Scratch:
• solve $x = \frac{3y - 2}{y - 1}$
• so $x \neq 3 \land y \neq 1$
Proof:
• define $g(x) = \frac{3x - 2}{x - 1}$
• observe $(\forall x \in \mathbb{R} \ \{1\})(g(x) \in \mathbb{R})$ because $x \neq 1$
• observe $g(x) \neq 3$ because $g(x)=3 \iff \frac{3x-2}{x-1} = 3 \iff 3x-2=3x-3 \iff -2=-3$
• so g is well-defined
• then $g(f(x)) = \frac{3\frac{3-2}{x-3}-2}{\frac{x-2}{x-3}-1} = x$ so $g \circ f = id_{\mathbb{R} \ \{3\}}$ holds
• show the same thing for $f(g(x))$
• f is invertible -> f is a bijection
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# Diophantine equation with three variables
The question is:
Nadir Airways offers three types of tickets on their Boston-New York flights. First-class tickets are \$140, second-class tickets are \$110, and stand-by tickets are \$78. If 69 passengers pay a total of$6548 for their tickets on a particular flight, how many of each type of ticket were sold?
Now I set up my equation as
$140x+110y+78z=6548$
But I'm confused how to go from here. I know I need to find the GCD in order to evaluate that the equation has a solution and then set up my formulas for $x=x_{0}+\frac{b}{d}(n)$ and $y=y_{0}-\frac{a}{d}(n)$
Ive solved Diophantine equations before but only in the form $ax+by=c$. How do I continue from here? I'm not interested in the solution, I can do that by myself, but I would like to know the process from solving these types of Diophantine equations.
• Also $x+y+z=69$. I hope that you can find it. Have good days – scarface Oct 8 '17 at 19:32
• @scarface thank you! I can't believe I missed that, I feel so embarrassed for not realizing that. – user482578 Oct 8 '17 at 19:38
• $(x,y,z)=(9,19,41)$ – Donald Splutterwit Oct 8 '17 at 19:50
• After considering the sum of the passenger you should get $$31 x+16 y=583$$ – Raffaele Oct 8 '17 at 19:58
$140x+110y+78z=6548$
and
$x + y + z = 69$
$\implies 78x + 78y + 78z = 69*78 = 5382$
$\implies 62x + 32y = 1166 \implies 31x + 16y = 69*78 = 583$
And we can quickly deduce that $x = 9, y = 19, z = 41$ (by simple inspection in my case - using that we only have integer values for $x,y,z$.
If the $\gcd$ of the ticket prices does not divide the total revenue, then you are correct that there will be no integer solution. However you are not immediately guaranteed a solution if the $\gcd$ does divide the revenue, because we are constrained to non-negative numbers of tickets. So we could potentially run into a Frobenius-coin-type failure.
Here the total number of tickets reduces this to a simple "two-coin" problem: | {
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Here the total number of tickets reduces this to a simple "two-coin" problem:
\begin{align} &&140x+110y+78z &= 6548\\ \text{divide by }\gcd(x,y,z)=2&& 70x+55y+39z &= 3274\\ &&x+y+z &= 69\\ \text{multiply by }39 && 39x+39y+39z &= 2691\\ \text{subtract eqns} && 31x+16y &= 583\\ \bmod 16 && 31x\equiv 15x \equiv -1x&\equiv 583\equiv 7\\ \bmod 16 && x&\equiv -7\equiv 9\\ \text{test }x=25 && 31\cdot25 &= 775>583 \\ \text{thus }x=9 && 31\cdot 9 +16y&= 583 \\ && y= (583-279)/16 &= 19\\ && z= 69-(19+9) &= 41\\ \end{align}
In the reduced equation $31x+16y = 583$, since $583>(31{-}1)\cdot (16{-}1)$ the coin problem issue could not apply - the total is big enough to guarantee a solution. | {
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# Math Help - Calculating position on a circumference
1. ## Calculating position on a circumference
Hello all, I hope someone is able to help me get my head round this little problem.
If I have a circle that is centered at (200,200) and its radius is 150, how do I calculate the point at any given angle? For example, I know that at 90 degrees, the point on the circumference will be (350,200), because I can calculate that manually, but what about more arbitrary degrees like 92.5 or 108?
Any help would be appreciated!
Thanks!
2. Originally Posted by gryphon5
Hello all, I hope someone is able to help me get my head round this little problem.
If I have a circle that is centered at (200,200) and its radius is 150, how do I calculate the point at any given angle? For example, I know that at 90 degrees, the point on the circumference will be (350,200), because I can calculate that manually, but what about more arbitrary degrees like 92.5 or 108?
Any help would be appreciated!
Thanks!
If the angle is 90 degrees, the point is (350,200)?
Do your angles start at the upper axis, the "North axis", then go clockwise?
If your angles start from the normal "East axis" then go counterclockwise, then the point at 90 degrees should be (200,350).
Whatever way you have there, for arbitrary angles/degrees, you just get the components of the 150-radius that are parallel to your axes.
Let us say your angles start the usual East-axis, or positive x-axis, and then go counterclockwise.
If the angle is 90 degrees,
x = 200 +150cos(90deg) = 200 +0 = 200
y = 200 +150sin(90deg) = 200 +150 = 350
Hence, point (200,350).
If the angle is 30 degrees,
x = 200 +150cos(30deg) = 200 +129.9 = 329.9
y = 200 +150sin(30deg) = 200 +75 = 275
Hence, point (329.9,275).
If the angle is 108 degrees,
x = 200 +150cos(108deg) = 200 -46.35 = 153.65
y = 200 +150sin(108deg) = 200 +142.66 = 342.66
Hence, point (153.65,342.66).
Etc....
3. Thanks vry much for your fast response. | {
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Etc....
3. Thanks vry much for your fast response.
Sorry for not including all the information, I should have stated...
This problem is for a computer application to draw an image, and since computers treat (0,0) as the top left of the screen (with the angles going clockwise), this is what I have used.
How would this change the examples you gave?
Thanks again!
4. Originally Posted by gryphon5
Thanks vry much for your fast response.
Sorry for not including all the information, I should have stated...
This problem is for a computer application to draw an image, and since computers treat (0,0) as the top left of the screen (with the angles going clockwise), this is what I have used.
How would this change the examples you gave?
Thanks again!
I see.
Then here is the change.
Let us call the vertical axis as y-axis also. The horizontal axis as x-axis also. The angles start from the upper or positive y-axis, going clockwise.
Our coordinates are in the usual (x,y) ordered pair.
If the angle is 30 degrees,
y = 200 +150cos(30deg) = 200 +129.9 = 329.9
x = 200 +150sin(30deg) = 200 +75 = 275
Hence, point (275,329.9).
If the angle is 108 degrees,
y = 200 +150cos(108deg) = 200 -46.35 = 153.65
x = 200 +150sin(108deg) = 200 +142.66 = 342.66
Hence, point (342.66,342.66).
If the angle is 92.5 degrees,
y = 200 +150cos(92.5deg) =200 -6.54 = 193.46
x = 200 +150sin(92.5deg) = 200 +149.86 = 349.86
Hence, point (349.86,193.46).
If the angle is 328.4 degrees,
y = 200 +150cos(328.4deg) = 200 +127.76 = 327.76
x = 200 +150sin(328.4deg) = 200 -78.60 = 121.40
Hence, point (121.40,327.76).
In other words,
For the x-component of the radius, use sine.
For the y-component of the radius, use cosine.
It's the reverse if you're doing them in the usual manner where the angles start from the positive x-axis, going counterclockwise.
5. Hello, gryphon5! | {
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5. Hello, gryphon5!
If I have a circle that is centered at (200,200) and its radius is 150,
how do I calculate the point at any given angle?
Code:
| * * *
| * * P
| * *
| * r / |*
| / |
| * / θ | *
| * O* - - - + *
| * (h,k) Q *
|
| * *
| * *
| * *
| * * *
|
- + - - - - - - - - - - - - - - -
|
Consider a circle with radius $r$ with center $O(h,k)$.
Point $P$ creates $\angle POQ$ with the horizontal.
In right triangle $PQO$, we have:
. . $\cos\theta = \frac{OQ}{r}\quad\Rightarrow\quad OQ = r\cos\theta$
. . $\sin\theta = \frac{PQ}{r}\quad\Rightarrow\quad PQ = r\sin\theta$
The $x$-coordinate of $P$ is: . $x \:=\:h + OQ\:=\:h + r\cos\theta$
The $y$-coordinate of $P$ is: . $y \:=\:k + PQ \:=\:k + r\sin\theta$
Therefore, point $P$ is at: . $\left(h + r\cos\theta,\:k + r\sin\theta\right)$
6. Wow, very helpful, thanks to you both that has made things much clearer
7. Sorry to re-open an old thread, but the markup seems to have gone weird, is it possible for someone to ressurect it so I can read the equations again?
Thanks,
gryphon
8. Hello!
I'll try to format all this without LaTeX . . .
Code:
| * * *
| * * P
| * *
| * r / |*
| / |
| * / θ | *
| * O* - - - + *
| * (h,k) Q *
|
| * *
| * *
| * *
| * * *
|
- + - - - - - - - - - - - - - - -
|
Consider a circle with radius r with center O(h,k).
Point P creates /POQ with the horizontal.
In right triangle PQO, we have:
. . cosθ = OQ/r . . OQ = r·cosθ
. . sinθ = PQ/r . . . PQ = r·sinθ
The x-coordinate of P is: .x .= .h + OQ .= .h + r·cosθ
The y-coordinate of P is: .y .= .k + PQ . = .k + r·sinθ
Therefore, point P is at: .(h + r·cosθ, k + r·sinθ) | {
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# Determining Linearly Dependent Vectors
I am learning about Linear dependent vector from here
But I am unable to grasp the following equation:
If no such scalars exist, then the vectors are to be linearly independent.
$$c_1\begin{bmatrix}x_{11}\\x_{21}\\\vdots\\x_{n1}\\ \end{bmatrix}+c_2\begin{bmatrix}x_{12}\\x_{22}\\\vdots\\x_{n2}\\ \end{bmatrix}+\cdots+c_n\begin{bmatrix}x_{1n}\\x_{2n}\\\vdots\\x_{nn}\\ \end{bmatrix}=\begin{bmatrix}0\\0\\\vdots\\0\\ \end{bmatrix}\\ \begin{bmatrix}x_{11}&x_{12}&\cdots&x_{1n}\\x_{21}&x_{22}&\cdots&x_{2n}\\ \vdots&\vdots&\ddots&\vdots\\x_{n1}&x_{n2}&\cdots&x_{nn}&\\ \end{bmatrix}\begin{bmatrix}c_1\\c_2\\\vdots\\c_n\end{bmatrix}=\begin{bmatrix}0\\0\\\vdots\\0\end{bmatrix}$$ In order for this matrix equation to have a nontrivial solution, the determinant must be $0$
How the first equation is reduced to the second one?
• What exactly are you asking? – user418131 Sep 18 '18 at 9:05
• How the first equation is reduced to the second one? – Cody Sep 18 '18 at 9:06
• Write a formula for the $i$-th entry of the vector above and below and you will see they are the same. – Michal Adamaszek Sep 18 '18 at 9:08
• By the use of matrix multiplication. – user418131 Sep 18 '18 at 9:08
• The first equation is actually equivalent to $n$ equations. Do you know how to interchange linear equations with an analogous matrix equation? – user418131 Sep 18 '18 at 9:10
$$\begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{bmatrix}\begin{bmatrix} c_{1} \\ c_{2}\end{bmatrix}=\begin{bmatrix} c_{1}x_{11}+c_2x_{12} \\ c_{1}x_{21}+c_2x_{22} \end{bmatrix}=c_1\begin{bmatrix} x_{11} \\ x_{21}\end{bmatrix}+ c_2\begin{bmatrix} x_{21} \\ x_{22}\end{bmatrix}$$
Notice that $c_1$ is only multiplied to entries in the first column and $c_2$ is only multiplied to the entries in the second column. | {
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This results from the definition of scalar multiplication and addition of matrices: \begin{align} c_1\begin{bmatrix} x_{11}\\x_{21}\\\vdots\\x_{n1} \end{bmatrix}+c_2\begin{bmatrix} x_{12}\\x_{22}\\\vdots\\x_{n2} \end{bmatrix}+\dots +c_n\begin{bmatrix} x_{1n}\\x_{2n}\\\vdots\\x_{nn} \end{bmatrix} &= \begin{bmatrix} c_1x_{11}\\c_1x_{21}\\\vdots\\c_1x_{n1} \end{bmatrix}+\begin{bmatrix} c_2x_{12}\\c_2x_{22}\\\vdots\\c_2x_{n2} \end{bmatrix}+\dots +\begin{bmatrix} c_nx_{1n}\\c_nx_{2n}\\\vdots\\c_nx_{nn} \end{bmatrix}\\[1ex] &= \begin{bmatrix} c_1x_{11}+c_2x_{12}+\dots+c_nx_{1n}\\c_1x_{21}+c_2x_{22}+\dots+c_nx_{2n}\\\dots\dots\dots\dots\dots\dots\dots\dots\\c_1x_{n1}+c_2x_{n2}+\dots+c_nx_{nn} \end{bmatrix} \end{align}
• So we are essentially writing a set of linear equations in matrix form, right? – Cody Sep 18 '18 at 9:54
• It's exactly that. – Bernard Sep 18 '18 at 10:00
Recall that the product $A\vec c$ can be interpreted as the linear combination of the colums $\vec x_i$ of $A$ by the coordinates $c_i$ of $\vec c$
$$A\vec c =\sum c_i\vec x_i$$
Refer also to the related | {
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# Plot Transfer Function Matlab | {
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Matlab also o ers w a ys to turn a sequence of graphs in toamo vie, con. You need to use the tf (link) function to produce a system object from your transfer function, and the lsim (link) function to do the simulation. Yes, i have Control System Toolbox. Bode Plot Example of First-Order System using Matlab. 2 in Control Systems By Nagoor Kani. Let me add to that last comment. Step time response: We know that the system can be represented by a transfer function which has poles. After reading the MATLAB control systems topic, you will able to solve problems based on the control system in MATLAB, and you will also understand how to write transfer function, and how to find step response, impulse response of various transfer systems. What does the MATLAB function ''tf2ss'' do ? Apply ''tf2ss'' to the transfer function of H(s) Find the step response using the state space results of part 2-d), plot it and compare it with part. I get the transfer function using. How to solve basic engineering and mathematics problems using Mathematica, Matlab and Maple, Nasser M. As a result this article presents an alternative that requires more lines of code but offers the full formatting flexibility of the generic plot command. a sensor with 0. Unformatted text preview: 6. A simple trick I found online was to use step() and divide the TF by s, and it should simulate a ramp response, step(G/s). Transfer function G(s) with plot or data. PI(D) Algorithm in MATLAB •We can use the pid() function in MATLAB •We can define the PI(D) transfer function using the tf() function in MATLAB •We can also define and implement a discrete. This studio will focus on analyzing the time response of linear systems represented by transfer function models. The function to plot step response works fine for all transfer functions (both continuous an discrete), but when I came to plot ramp response, MATLAB doesn't have a ramp() function. We can see that the PID controller we designed works well in the face of | {
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have a ramp() function. We can see that the PID controller we designed works well in the face of uncertainty in estimated transfer function parameters. Yes, i have Control System Toolbox. I want the graph to start at 5 after it leaves the transfer function block in Simulink. The name of the file and of the function should be the same. A SISO continuous-time transfer function is expressed as the ratio:. Hence, x-axis in your plot will only signify the total number of data points in FF_mag_nw. Numerator or cell of numerators. Plot Bode asymptote from Transfer Function. This plotting script employs the function cal_avg. It is obtained by applying a Laplace transform to the differential equations describing system dynamics, assuming zero initial conditions. Transfer functions can be used to represent closed-loop as well as open-loop systems. The first parameter is a row vector of the numerator coefficients. This shows how to use Matlab to solve standard engineering problems which involves solving a standard second order ODE. t is the time, ranging from 0 seconds to 10 seconds and w is a pulsation of 1. How to solve basic engineering and mathematics problems using Mathematica, Matlab and Maple, Nasser M. Transfer function G(s) with plot or data. The optical transfer function is not only useful for the design of optical system, it is also valuable to characterize manufactured systems. Add these time functions to produce the output. Plot transfer function response. rlocus(sys) calculates and plots the root locus of the open-loop SISO model sys. If needed, you can then convert the identified state-s[ace model into a transfer function using tf. The transfer function was $$\frac{20000}{s+20000}$$. MATLAB: A Practical Introduction to Programming and Problem Solving, winner of TAA’s 2017 Textbook Excellence Award ("Texty"), guides the reader through both programming and built-in functions to easily exploit MATLAB's extensive capabilities for tackling engineering and scientific | {
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functions to easily exploit MATLAB's extensive capabilities for tackling engineering and scientific problems. Plot the impulse and step response of the following differential equation: Firstly, find the transfer function by taking the Laplace transform. Running this m-file in the Matlab command window should gives you the following plot. The sys (system) structure in MATLAB v5 is very powerful, and it allows you to form complicated systems by joining together simpler systems. When a single vector argument is passed to plot, the elements of the vector form the dependent data and the index of the elements form the dependent data. If sys is a multi-input, multi-output (MIMO) model, then bode produces an array of Bode plots, each plot showing the frequency response of one I/O pair. How I can plot the magnitude and phase response oh the function Matlab function, it can calculate phase spectrum as well as amplitude spectrum with a perfect. how find ramp response. By applying Cauchy’s principle of argument to the open-loopsystem transfer function, we will get information about stability of the closed-loopsystem transfer function and arrive at the Nyquist stability criterion (Nyquist, 1932). A = logsig(N,FP) takes N and optional function parameters,. Title: 3D Plot Transfer Function Author: J. ( iddata or idfrd) where I gona used tfest function to estimate d transfer function. Function Plotting in Matlab. Note that the system transfer function is a complex function. Question: 9. Plot pole-zero diagram for a given tran. The Matlab function freqz also uses this method when possible ( e. RLocusGui is a graphical user interface written in the Matlab® programming language. (c) Clicking on the pole at 3/2 + √ 15/2 we see that Matlab predicts overshoot of 9. Hello, i am trying to make a bode plot of the transfer function of a twin-t notch filter, that i am analyzing. H(s) is a complex function and 's' is a complex variable. it has an amplitude and a phase, and ejωt=cosωt+jsinωt. | {
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function and 's' is a complex variable. it has an amplitude and a phase, and ejωt=cosωt+jsinωt. Creates a continuous-time transfer function with numerator and denomi-nator specified by num and den. Control System Toolbox™ software supports transfer functions that are continuous-time or discrete-time, and SISO or MIMO. The transfer function is T s =. So the problem is how to run a Simulink model. More and more MATLAB users are using automation servers as part of continuous integration workflows. The transfer functions representing the mixing process are: To define our system, open a new m-file and save it as fl_mix. This studio will focus on analyzing the time response of linear systems represented by transfer function models. time response of a second order system 7. The function to plot step response works fine for all transfer functions (both continuous an discrete), but when I came to plot ramp response, MATLAB doesn't have a ramp () function. Frequency response plots show the complex values of a transfer function as a function of frequency. Then, you can apply any signal to the block and it will give you the output. ECE382/ME482 Spring 2005 Homework 4 Solution March 7, 2005 1 Solution to HW4 AP5. Question: 9. The root locus of an (open-loop) transfer function is a plot of the locations (locus) of all possible closed-loop poles with some parameter, often a proportional gain , varied between 0 and. i want write a script to plot a graph for the transfer function [H(f)] for a band pass filter, |H(f)| against frequency and the phase of H(f) (degrees) against frequency, im very new to matlab so the syntax is not 100%, im getting confused because everything is auto formatted in matrix form. The transfer function of a certain fourth-order, low pass, inverse Chebyshev filter with 3 dB frequency at 9600 radians/second will be used in all examples. Bode Plot Definition H. using % a) standard plotting and complex number capabilities, % b) standard plotting and complex number | {
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% a) standard plotting and complex number capabilities, % b) standard plotting and complex number capabilities for generating Bode plots, and % c) built in Bode plot function. Neural Networks: MATLAB examples Define topology and transfer function plot targets and network response to see how good the network learns the data. If you want a different type of plot, look under Edit:Plot Configurations. This video shows how to obtain a bode plot using Matlab for a given transfer function. For example, consider the transfer function. And could tfest gives the transfer function where the data is in decibel. This function has three poles, two of which are negative integers and one of which is zero. There is a program within Matlab called Simulink. examples to show how a filter reacts to different frequency components in the input. A simple trick I found online was to use step() and divide the TF by s and it should simulate a ramp response, step(G/s). We can define the function having a scalar number as an input. 528 and no lag compensator, the. Make sure to "turn off" the feedback loop by setting the value of the gain to equal zero. You can add a controller, and compute the closed-loop transfer function. Run the simulation • Set the simulation to run for 30 seconds: Simulation->Configuration Parameters. sys_p is an identified transfer function model. It seems to me that the standard way of plotting the frequency response of the filter is to use a Bode plot. Add these time functions to produce the output. To construct a Bode plot from a transfer function, we use the following command:. I get the transfer function using. Plot the frequency spectrum (i. This tutorial discusses some of the different ways that MATLAB and Simulink interact. 5 (R2007b)] [Book]. Lattice or lattice ladder to transfer function. A SISO continuous-time transfer function is expressed as the ratio:. H is just the way to call what is the 'transfer matrix' of my system. on the plot and as thoroughly as you | {
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the way to call what is the 'transfer matrix' of my system. on the plot and as thoroughly as you can, on the similarities and differences, if any, to the low–gain system step response from questions 8–9. In engineering, a transfer function (also known as system function or network function) of an electronic or control system component is a mathematical function which theoretically models the device's output for each possible input. Using MATLAB, input the transfer function H(s) H(s) = [ 23+3 s^2+2s+1]^T/( s^2+0. Transfer functions calculate a layer’s output from its net input. The figure produced by the bode(sys) function can be copied and pasted into wordprocessors and other programs. The plot displays the magnitude (in dB) and phase (in degrees) of the system response as a function of frequency. As a general rule, matlab programs should avoid iterating over individual samples whenever possible. Use the standard deviation data to create a 3σ plot corresponding to the confidence region. We are going to develop a function that will return the voltage and corresponding time of the response. A popular option is Jenkins. Using Matlab to create Transfer functions and bode plots. Let me add to that last comment. The optical transfer function is defined as the Fourier transform of the impulse response of the optical system, also called the point spread function. H is just the way to call what is the 'transfer matrix' of my system. Several examples of the construction of Bode Plots are included in this file. This MATLAB function computes the inverse Fast Fourier Transform of the optical transfer function (OTF) and creates a point-spread function (PSF), centered at the origin. sys = tf(B,A); t = 1:length(u); y = lsim(sys,u,t); figure plot(y) I am sure the estimated transfer function is correct, since it resembles the original system so much. Every digital filter can be specified by its poles and zeros (together with a gain factor). The roots of a(s) are called poles of the | {
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by its poles and zeros (together with a gain factor). The roots of a(s) are called poles of the system. The frequency response of a system, is just the transfer function, evaluated at. Frequency Response, Bode Plots, and Resonance 3. You can multiply transfer functions sys1=tf(num1,den1) and sys2 = tf(num2, den2) using sys3=sys1*sys2. So the problem is how to run a Simulink model. A simple trick I found online was to use step() and divide the TF by s and it should simulate a ramp response, step(G/s). The result shown in the command window is ‘tf = empty transfer function’. A Bode plot is a graphical representation of a linear, time-invariant system transfer function. Bode introduced a method to present the information of a polar plot of a transfer function GH(s), actually the frequency response GH (jω), as two plots with the angular frequency were at the common axis. transfer function Eq. Click on the transfer function in the table below to jump to that example. How do I plot the contour of a given Nyquist plot onto the *s-plane of a given transfer function on Matlab? *s-plane: is the complex plane on which Laplace transforms are graphed. orF the original control system with K = 1. , RCL circuit with voltage across the capacitor C) as the output) is where is an arbitrary gain factor. The transfer function for the time delay can not be directly represented in MATLAB. Another thing is MATLAB plots infinity as one. More and more MATLAB users are using automation servers as part of continuous integration workflows. Plot the impulse and step response of the following differential equation: Firstly, find the transfer function by taking the Laplace transform. Yes, i have Control System Toolbox. My simulink model contain a bunch of 1/z unit delays, sums and gains. Transfer Function Representations. Of course you can, and T is called time delay. transfer function based on your choices, and compare the rise time, overshoot and damped oscillation frequency of the response you | {
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choices, and compare the rise time, overshoot and damped oscillation frequency of the response you get from MATLAB with the corresponding values that you expect from the theory. Numerator or cell of numerators. Two transfer functions are combined to create a plant model. form the complete transfer function with the lag compensator added in series to th original system; plot the new Bode plot and determine phase margin and observe that it is the required phase margin; Now to do this In Matlab let us take a question. A transfer function is a convenient way to represent a linear, time-invariant system in terms of its input-output relationship. Calculate poles and zeros from a given transfer function. It is obtained by applying a Laplace transform to the differential equations describing system dynamics, assuming zero initial conditions. Bode Plot Definition H. The above plot shows that. In addition to estimating continuous-time transfer functions, System Identification Toolbox lets you estimate continuous-time state-space models and process models (special, low-order. The figure below shows a unity-feedback architecture, but the procedure is identical for any open-loop transfer function , even if some elements of the. Enter transfer function in MATLAB. Bode Plot Example of First-Order System using Matlab. Example1: Let us plot the Bode Plot for each transfer function and in doing so we will see the added functionality that can be achieved from the Bode plot function in MATLAB. How to make a plot with logarithmic axes in MATLAB ®. Plot of the disturbance model, called noise spectrum. purelin is a neural transfer function. Functions operate on variables within their own workspace, which is also called the local workspace, separate from the workspace you access at the MATLAB command prompt which is called the base workspace. Creating Bode Plots. This is achieved using the MATLAB-Simulink API (application program interface) commands. The function for step response works | {
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MATLAB-Simulink API (application program interface) commands. The function for step response works fine for all transfer functions (both continuous an discrete), but when I came to ramp response, MATLAB doesn't have a ramp() function. For MIMO models, pzmap displays all system poles and transmission zeros on a single plot. 3 MAXIMUM POWER TRANSFER 4. rlocus(sys) calculates and plots the root locus of the open-loop SISO model sys. I have done the calculations manually using Euler's formula, but now the assignment is asking me to compare these plots with the plots using freqz in MATLAB. Try this in matlab: s = tf('s'); T = 1; G = 1/s; Gd = exp(-s*T)/s; bode(G,Gd) and it will yield the following Bode diagram:. State space to transfer function. Plot the frequency spectrum (i. I've been trying to practice using Matlab for circuit analysis and am trying to create a transfer function plot of a high pass filter where the gain is in volts/volts not in dB. This way you can easily see how the two functions are similar or different from each other. So basically I have a digital filter and I need to plot a transfer function of this filter. This function creates arrows that go out from the origin of the axes in a polar coordinate system. logsig is a transfer function. The default formatting of most MATLAB plots is good for analysis but less than ideal for dropping into Word and PowerPoint documents or even this website. However, this would execute much slower because the matlab language is interpreted, while built-in functions such as filter are pre-compiled C modules. Estimating Other Model Types. Once plotted, you will. A SISO continuous-time transfer function is expressed as the ratio:. The transfer function generalizes this notion to allow a broader class of input signals besides periodic ones. 1 1]); >> grid >> grid The grids are optional. on the plot and as thoroughly as you can, on the similarities and differences, if any, to the low–gain system step response from questions | {
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can, on the similarities and differences, if any, to the low–gain system step response from questions 8–9. Also how to plot points on the bode plots and how to find help in Matlab. The first two software packages are free alternatives to Matlab, and their use is encouraged. A = purelin(N,FP) takes N and optional function parameters,. You can use static gain transfer function model sys1 obtained above to cascade it with another transfer function model. The roots of a(s) are called poles of the system. % Transfer function: 2500(10 + jw). RLocusGui is a graphical user interface written in the Matlab® programming language. MATLAB's tfestimate will produce a numerical estimate of the magnitude and phase of a transfer function given an input signal, an output signal, and possibly other information. Question: 9. and your transfer function is : 𝑉 𝑉𝑖 = 2 1+ 2 Now use the coefficients in MATLAB to plot the frequency response of this analog filter. Transfer functions calculate a layer’s output from its net input. CheungSlide 12. Transfer Function in MATLAB: As noted previously that the transfer function represents the input and output of the system in terms of the complex frequency variable so that the transfer function can give the complete information about the frequency response of the system. Then as a function of ω, the radian frequency, you plot the real and Imaginary parts from ω=0 to ω=∞. The transfer function is T s =. For example, consider the transfer function. Matlab also o ers w a ys to turn a sequence of graphs in toamo vie, con. Transfer functions calculate a layer's output from its net input. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. You should get something like the following: To input this into MATLAB, choose some values for the resistor, capacitor and inductor constants: % Values for constants. There is a | {
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some values for the resistor, capacitor and inductor constants: % Values for constants. There is a function bodeplot in Matlab which for instance takes an argument calculated with tf, which in turn takes a numerator and denominator. 1s + 1)(s+1)(10s+1). By applying Cauchy’s principle of argument to the open-loopsystem transfer function, we will get information about stability of the closed-loopsystem transfer function and arrive at the Nyquist stability criterion (Nyquist, 1932). zeros and poles from transfer function 3. Note that Eq. Use the standard deviation data to create a 3σ plot corresponding to the confidence region. % Transfer function: 2500(10 + jw). The root locus of an (open-loop) transfer function is a plot of the locations (locus) of all possible closed-loop poles with some parameter, often a proportional gain , varied between 0 and. ramp response of a transfer function 6. I have never in my whole life heard of a 3D transfer function, it doesn't make sense. The above command will plot FF_mag_nw which is a one-dimensional matrix against its row number as its x co-ordinate. To learn more about the use of functions, go through the user guide. The function depends on real input parameters. A SISO continuous-time transfer function is expressed as the ratio:. % There are some sample functions below that can be copied and pasted into the % proper location. Zeros, Poles and Pole–Zero Map of a Transfer Function The command tf2zp is used to obtain the zeros z, poles p and gain k of the transfer function … - Selection from MATLAB® and Its Applications in Engineering: [Based on MATLAB 7. From MATLAB command window, we will call the function CreatePlant to create the transfer function mentioned in shown: sys=CreatePlant(1,[1000 300 30 1]); step(sys) b. To plot more than one transfer function use the following syntax: bode(sys1,sys2,…). What you have to do is evaluate the transfer function's Real and Imaginary parts. So the problem is how to run a Simulink model. | {
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the transfer function's Real and Imaginary parts. So the problem is how to run a Simulink model. This way you can easily see how the two functions are similar or different from each other. Bode plot diagram state space. How to invert a transfer function in simulink (matlab)? I am having disturbance at the out put of my plant. In order to draw Bode Plot, we need transfer function from which we deduce the equations for Magnitude and Phase. transfer function based on your choices, and compare the rise time, overshoot and damped oscillation frequency of the response you get from MATLAB with the corresponding values that you expect from the theory. %simulate the estimated transfer function. Functions operate on variables within their own workspace, which is also called the local workspace, separate from the workspace you access at the MATLAB command prompt which is called the base workspace. Once you have converted your differential equation to a transfer function you can then use the options within the simulink control space to input your transfer function. Hello, lets say I have an image then I adjusted the contrast by using histogram equalization (histeq) Is there a way to plot or get an image of the transfer function that this command uses ?. not both the magnitude and and the phase. Then, you can apply any signal to the block and it will give you the output. It then shows how to generate the Bode plots and the step response plots of the transfer function. MATLAB provides command for working with transforms, such as the Laplace and Fourier transforms. I think you are completely wrong: z does not represent a complex number, but the fact that your transfer function is a discrete one, rather than a continuous one (see the Z transform for more details). So basically I have a digital filter and I need to plot a transfer function of this filter. Singular values plot of a transfer function. We can define the function having a scalar number as an input. I will be solving the | {
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function. We can define the function having a scalar number as an input. I will be solving the question number 6. I want to have an equivalent input disturbance for the same. I've been trying to practice using Matlab for circuit analysis and am trying to create a transfer function plot of a high pass filter where the gain is in volts/volts not in dB. The locus of the roots of the characteristic equation of. Compare it to this, you want to plot a sine wave: x = sin(w*t), I hope you can agree with me that you cannot plot such a function (including axes) unless I specifically say e. purelin is a neural transfer function. Hello, lets say I have an image then I adjusted the contrast by using histogram equalization (histeq) Is there a way to plot or get an image of the transfer function that this command uses ?. Transfer functions are a frequenc view the full answer. State space controlability and observability. 1 GRAPH FUNCTIONS 2. System Stability If a linear system is described by a transfer function H(s), the system is said to be stable if. To export the linearized system to the Workspace so you can use it with other design tools in Matlab, select File: Export. I have never in my whole life heard of a 3D transfer function, it doesn't make sense. Transfer functions calculate a layer’s output from its net input. How can I plot two functions in the same graph?. I wanted to know how I can go about plotting a simple bode magnitude transfer function in LaTeX. Use the Laplace transform to solve for the time response and MATLAB for calculation and plotting. orF the original control system with K = 1. Singular values plot of a transfer function. That found the transfer function for the RL series circuit with a 20K resistor and a Inductor with a value of 500mH. The transfer function of the LTI system is the ratio of Laplace transform of output to the Laplace transform of input of the system by assuming all the initial conditions are zero. Bode Plot with Magnitude on a dB Scale | {
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the system by assuming all the initial conditions are zero. Bode Plot with Magnitude on a dB Scale in MATLAB % Magnitude of a Transfer Function on a dB Plot % Save output figures in bitmap mode for best quality. Hello, i am trying to make a bode plot of the transfer function of a twin-t notch filter, that i am analyzing. Abbasi [ next ] [ prev ] [ prev-tail ] [ tail ] [ up ] 1. 11/12/18 9 Oblique Wing Concepts •High-speed benefitsof wing sweep without the heavy structure and complex mechanism required for symmetric sweep •Blohmund Voss,R. You can also do the evaluation for negative values of ω, remembering that in the complex domain the point at infinity is a single point. The transfer function of a position control. If sys has transfer function. In this video I will give you a very quick but needed description of how to plot Step Response of Transfer Function Using Matlab. it has an amplitude and a phase, and ejωt=cosωt+ jsinωt. MATLAB proved very capable at taking the Bode plot of a given transfer function using the online documentation. Transfer functions calculate a layer’s output from its net input. I want to have an equivalent input disturbance for the same. Use the Laplace transform to solve for the time response and MATLAB for calculation and plotting. Of course in interpreting the Bode plot of an unknown system, one is seeing the plot of the entire system, and one must pick out the components from the whole. Here we will learn how to write a Matlab code for creating a transfer function and then analyzing this transfer code for its reaction to several. Without defined values of R and C you won't get any transfer function. The s is jw. Enter transfer function in MATLAB. Transfer Function Matlab Example. This function can be applied to any of the following negative feedback loops by setting sys appropriately. Open loop system response Figure 3 Open loop system To plot the open loop response, perform the following steps: a. This is achieved using the | {
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system To plot the open loop response, perform the following steps: a. This is achieved using the MATLAB-Simulink API (application program interface) commands. it has an amplitude and a phase, and ejωt=cosωt+jsinωt. Plot the bode diagram for the transfer function and: Plot the Bode Diagrams Find the Gain and Phase margin For what value of K, the closed-loop system is unstable? Plot the Nyquist plot by using MATLAB Repeat part (3). Estimating Other Model Types. Examine the Root Locus Diagram of the following transfer function. H is just the way to call what is the 'transfer matrix' of my system. to create s as a variable and then use s in a line of code to make a transfer function. MATLAB Answers. Plot the frequency spectrum (i. For example, consider the transfer function. Frequency response is usually a complex valued function, so it can be written as , where is the magnitude response and is the phase response. This means that the characteristic equation of the closed loop transfer function has no zeros in the right half plane (the closed loop transfer function has no poles there). The step function is one of most useful functions in MATLAB for control design. This plot is the same as a Bode plot of the model response, but it shows the output power spectrum of the noise model instead. Enter transfer function in MATLAB. 9 s + 6538 It gives us the transfer function for everything except the capacitor and my question is how do I convert this transfer function into its equvalent R and L. You need to use the tf (link) function to produce a system object from your transfer function, and the lsim (link) function to do the simulation. To construct a Bode plot from a transfer function, we use the following command:. Introduction. PI(D) Algorithm in MATLAB •We can use the pid() function in MATLAB •We can define the PI(D) transfer function using the tf() function in MATLAB •We can also define and implement a discrete. The transfer function generalizes this notion to allow a | {
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can also define and implement a discrete. The transfer function generalizes this notion to allow a broader class of input signals besides periodic ones. Transfer functions calculate a layer’s output from its net input. This function is a modified version of the nyquist command, and has all the same attributes as the original, with a few improvements. bode automatically determines frequencies to plot based on system dynamics. Hello, lets say I have an image then I adjusted the contrast by using histogram equalization (histeq) Is there a way to plot or get an image of the transfer function that this command uses ?. When you call this function, you can specify system order as a vector, say [1 10], and the function will then return a plot helping you choose the best order as shown here. All the signals are transfer functions on the block diagrams. Plot Step Response of Transfer Function Using Simulink on Matlab. Root Locus with Time Delays. The plot function can accept one, two, or more arguments and produces a plot of the data contained in the arguments. Running this m-file in the Matlab command window should gives you the following plot. % You must edit this file under "** THE EQUATION: **" and enter the function y(s). Plot Bode asymptote from Transfer Function. A simple trick I found online was to use step() and divide the TF by s and it should simulate a ramp response, step(G/s). The Matlab Code: % This function creates two bode plots (amplitude and phase) for a transfer function. We can see that the PID controller we designed works well in the face of uncertainty in estimated transfer function parameters. Hello, i am trying to make a bode plot of the transfer function of a twin-t notch filter, that i am analyzing. % There are some sample functions below that can be copied and pasted into the % proper location. h = subplot(m,n,p), or subplot(mnp) breaks the Figure window into an m-by-n matrix of small axes, selects the pth axes object for for the current plot, and | {
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into an m-by-n matrix of small axes, selects the pth axes object for for the current plot, and returns the axis handle. The aperiodic pulse shown below: has a Fourier transform: X(jf)=4sinc(4πf) As shown in MATLAB Tutorial #2, we can plot the amplitude and phase spectrum of this signal. My First problem is actually inputting the transfer function into Matlab, It's the transfer function of a first order hold which is: [(1 + sT) / T] x [(1 - e^-sT) / s] x [(1 - e^-sT) / s]. frequency - y-axiith 20is is the 20•l f th it d f th t flog of the magnitude of the transfer function in dB and x-axis is ω - xx -axis isaxis is ωorfor f inlogscalein log scale dB log(f) or log(ω) EE40 Fall 2009 Prof. Transfer functions calculate a layer's output from its net input. In this example, we will draw two graphs with the same function, but in second time, we will reduce the value of increment. Hence, x-axis in your plot will only signify the total number of data points in FF_mag_nw. Such plots are known as pole-zero plots. magnitude of a step input. MATLAB plotting commands, you should become familiar with the following commands: • tf - This command is used to enter transfer functions. 5 (R2007b)] [Book]. This function creates arrows that go out from the origin of the axes in a polar coordinate system. A transfer function is represented by 'H(s)'. transfer function and impulse response are only used in LTI systems. Ask Question Can I just find the frequency gain and then use SVD in Octave/MATLAB to plot every dot ?. Any advance for the correct way to use ‘tfest’. Transfer Function Analysis and Design Tools. Abbasi [ next ] [ prev ] [ prev-tail ] [ tail ] [ up ] 1. Write out your answer for the H(s). This is achieved using the MATLAB-Simulink API (application program interface) commands. , RCL circuit with voltage across the capacitor C) as the output) is where is an arbitrary gain factor. State space controlability and observability. If you want a different type of plot, look under | {
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State space controlability and observability. If you want a different type of plot, look under Edit:Plot Configurations. There are two bode plots, one plotting the magnitude (or gain) versus frequency (Bode Magnitude plot) and another plotting the phase versus frequency (Bode Phase plot). Transfer Function Representations. | {
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# For any two sets, $A - B = B - A$ implies $A = B$
Is the following statement True or False:
For any two sets $A$ and $B$: If $A - B = B - A$ then $A = B$.
If it is true, prove it, otherwise provide a counterexample.
I am unable to come up with a counter example. I think the statement is true but how do I prove it?
• Suppose $x \in A - B$. Then $x \in B-A$. In particular $x \in B$, contradiction. So $A - B = \emptyset$ i.e. $A \subset B$. Same argument shows $B - A = \emptyset$ i.e. $B \subset A$. – hunter Jun 23 '17 at 6:55
If $A-B=B-A$ then for any $x\in A-B=B-A$ we $x\in A;x\in B; x\not \in A; x\not \in B$. That's a contradiction so $A-B=B-A$ is empty.
Thus there are no elements in $A$ that are not in $B$. In other words $A$ is a subset of $B$. Likewise there are no elements of $B$ that are in $A$. So $B$ is a subset of $A$.
So $A=B$.
If $A \setminus B = B \setminus A$, then
$A=A \setminus B \cup (A\cap B)= B \setminus A \cup (B \cap A) = B$.
Let’s use some Boolean algebra, in order to show a different point of view.
Let $C=A\cup B$; for a subset $X$ of $C$, denote $X^c=C\setminus X$; thus $$A\setminus B=A\cap B^c,\qquad B\setminus A=B\cap A^c=A^c\cap B$$ Then \begin{align} A&=A\cap C && \text{because $A\subseteq C$} \\ &=A\cap (B\cup B^c) && \text{because $C=B\cup B^c$} \\ &=(A\cap B)\cup(A\cap B^c) && \text{distributivity} \\ &=(A\cap B)\cup(A^c\cap B) && \text{hypothesis} \\ &=(A\cup A^c)\cap B && \text{distributivity} \\ &=C\cap B && \text{because $A\cup A^c=C$} \\ &=B && \text{because $B\subseteq C$} \end{align}
You also have \begin{align} A\cap B^c &=(A\cap B^c)\cap(B\cap A^c) && \text{hypothesis} \\ &=A\cap(B^c\cap(B\cap A^c)) && \text{associativity} \\ &=A\cap((B^c\cap B)\cap A^c) && \text{associativity} \\ &=A\cap(\emptyset\cap A^c) && \text{because $B\cap B^c=\emptyset$} \\ &=A\cap\emptyset \\ &=\emptyset \end{align} | {
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# Math Help - Rhombus Problem
1. ## Rhombus Problem
Given Rhombus ABCD (not shown) AB = 10 and AC = 12. Find AD and BD
I know that AD = 10 because the sides of a rhombus are all congruent. I cannot find what BD equals though. I thought it was 12 but I don't think the diagonals of a rhombus are congruent.
2. ## Re: Rhombus Problem
Originally Posted by Cake
Given Rhombus ABCD (not shown) AB = 10 and AC = 12. Find AD and BD
I know that AD = 10 because the sides of a rhombus are all congruent. I cannot find what BD equals though. I thought it was 12 but I don't think the diagonals of a rhombus are congruent.
$x^2+y^2=10^2=100$
$(10-x)^2+y^2=12^2=144$
$(10+x)^2+y^2=BD^2$
solving the first two equations we get $x=\frac{14}{5} and y=\frac{48}{5}$
plugging into the 3rd equation we get
$\left(10+\frac{14}{5} \right)^2+\left(\frac{48}{5} \right)^2=BD^2$
$\left(\frac{64}{5}\right)^2+\left(\frac{48}{5} \right)^2=\frac{6400}{25}=BD^2\Rightarrow BD=\frac{80}{5}=16$
3. ## Re: Rhombus Problem
Originally Posted by romsek
$x^2+y^2=10^2=100$
$(10-x)^2+y^2=12^2=144$
$(10+x)^2+y^2=BD^2$
solving the first two equations we get $x=\frac{14}{5} and y=\frac{48}{5}$
plugging into the 3rd equation we get
$\left(10+\frac{14}{5} \right)^2+\left(\frac{48}{5} \right)^2=BD^2$
$\left(\frac{64}{5}\right)^2+\left(\frac{48}{5} \right)^2=\frac{6400}{25}=BD^2\Rightarrow BD=\frac{80}{5}=16$
Holy cow! That's some math work! Thank you, is there any shorter way than this?
4. ## Re: Rhombus Problem
Originally Posted by Cake
Holy cow! That's some math work! Thank you, is there any shorter way than this?
Yes. The diagonals of a rhombus bisect each other at right angles.AB =10 AC =12. Note that there are four congruent triangles formed by them.1/2 of AC =6. 1/2 of BD=8 5-4- 3 right triangle
5. ## Re: Rhombus Problem | {
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5. ## Re: Rhombus Problem
Originally Posted by bjhopper
Yes. The diagonals of a rhombus bisect each other at right angles.AB =10 AC =12. Note that there are four congruent triangles formed by them.1/2 of AC =6. 1/2 of BD=8 5-4- 3 right triangle
Right, so half of AC is 6. For example, if we had point E in the middle of the rhombus. AE would be 6 because it is half of 12. then AB is 10. You just don't add 10 and 6 to get 16, right?
6. ## Re: Rhombus Problem
Originally Posted by Cake
Right, so half of AC is 6. For example, if we had point E in the middle of the rhombus. AE would be 6 because it is half of 12. then AB is 10. You just don't add 10 and 6 to get 16, right?
hrm I should have seen this.
No what you do is notice that
$\left(\frac{BD}{2}\right)^2+\left(\frac{12}{2} \right)^2=10^2$
$\left(\frac{BD}{2}\right)^2=100-36=64 \Rightarrow \frac{BD}{2}=8 \Rightarrow BD=16$
7. ## Re: Rhombus Problem
Hello, Cake!
$\text{Given rhombus }ABCD,\;AB = 10\text{ and }AC = 12.\;\text{ Find }AD\text{ and }BD.$
I know that AD = 10 because the sides of a rhombus are all congruent.
I cannot find what BD equals though.
I thought it was 12, but I don't think the diagonals of a rhombus are congruent.
If they were, you'd have a square.
The diagonals of a rhombus are perpendicular and bisect each other.
Hence: . $AO = OC = 6.$
Code:
A 10 B
o---------------o
/ * * /
/ *6 * /
/ * * /
10 / o /
/ * O* /
/ * *6 /
/ * * /
o---------------o
D C
In right triangle $AOB$, we find that $OB = 8.$
Therefore: . $BD = 16.$
8. ## Re: Rhombus Problem
Originally Posted by Soroban
Hello, Cake!
The diagonals of a rhombus are perpendicular and bisect each other.
Hence: . $AO = OC = 6.$ | {
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The diagonals of a rhombus are perpendicular and bisect each other.
Hence: . $AO = OC = 6.$
Code:
A 10 B
o---------------o
/ * * /
/ *6 * /
/ * * /
10 / o /
/ * O* /
/ * *6 /
/ * * /
o---------------o
D C
In right triangle $AOB$, we find that $OB = 8.$
Therefore: . $BD = 16.$
Makes sense ^_____^
Thank you guys! | {
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Categorías | {
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# Solve this functional equation:
Functional equations such as this one appear only once every several years on exams, so I feel it's hard to have a sure-fire way to approach the problem, unlike, say, solving a series convergence problem, multiple variable integration, or proving some results using basic Fourier series.
So, when I do see a solution offered for one of these problems and study the solution for a substantial amount of time, I still cannot remember how to solve these types of problems, when I come across another one.
But the question is:
Find all the real-valued continuous functions $f$ on $\mathbb R$ which satisfy $$f(x)f(y)=f(x_1)f(y_1)$$
for all $x$, $y$, $x_1$, $y_1$ such that $x^2+y^2=x_1^2+y_1^2$.
Ideally, besides offering a solution, I would love to hear about your intuition on how to solve these functional equations.
Thanks,
• +1. I wanna know how the experts solve functional equations too.... – Jack's wasted life Jun 26 '15 at 22:56
You can linearize the problem by introducing $$g(u):=\ln\left(f(\sqrt u)\right).$$
Then with $u=x^2,v=y^2$, $$u+v=u'+v'\implies g(u)+g(v)=g(u')+g(v').$$
Setting $u'=0,v'=u+v$,
$$g(u)+g(v)=g(0)+g(u+v)$$
shows that the function must be affine,
$$g(u)=au+b,$$ and $$f(x)=e^{ax^2+b}=F_0\left(\frac{F_1}{F_0}\right)^{x^2}.$$
The intuitions/tricks behind this: | {
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The intuitions/tricks behind this:
• it is often advantageous to linearize to benefit of what we know from linear algebra and make the equations look more familiar;
• products can be linearized by means of logarithms;
• non-linear functions can be linearized by means of a change of variable with the function inverse;
• when you have a property involving several variables, try to exploit it by assigning particular values to some of them.
• Hi @YvesDaoust - I really like this approach. I just have one follow-up question: how do you get the line g(u) = au+b, hence showing the function, g, is affine? – User001 Jun 29 '15 at 2:03
• From your previous line, it follows that g(u) = g(0) + g(u+v) - g(v)... – User001 Jun 29 '15 at 2:06
• @LebronJames: let $h(u)=g(u)-g(0)$, then $h(u)+h(v)=h(u+v)$. Assuming $h$ continuous, this is enough to say that $h$ is linear. – Yves Daoust Jun 29 '15 at 6:19
All solutions are the functions $f(x) = \alpha e^{\beta x^2}$, $\alpha,\beta \in \mathbb{R}$. Any of this functions satisfies the OP query: $$f(x)f(y) = \alpha e^{\beta x^2} \alpha e^{\beta y^2} = \alpha^2 e^{\beta(x^2 + y^2)} = \alpha^2 e^{\beta(x_1^2 + y_1^2)} = \alpha e^{\beta x_1^2} \alpha e^{\beta y_1^2} = f(x_1)f(y_1) \, .$$ | {
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Here is why these are the only ones. Observe that the hypothesis implies that there is a function $\psi$ such that $$f(x)f(y) = \psi(x^2 + y^2) \, .$$ If $f(0) = 0$ then $0.f(y) = \psi(y^2) = 0$ hence $f(x).f(y) \equiv 0$. So $f$ must be identically zero. Assume that $f(0) = \alpha \neq 0$. Then $\tilde{f}(x) := \frac{f(x)}{\alpha}$ also satisfies the OP hypotesis. So we can assume w.l.o.g. that $\alpha = 1$. From this we get that $f(x) = \psi(x^2)$ and $\psi(r)$ is a continuous function for $r \geq 0$. Moreover the function $\psi$ satisfies $$f(x) f(y) = \psi(x^2) \psi(y^2) = \psi(x^2 + y^2) \, .$$ By taking $x=y$ we see that $\psi \geq 0$. Actually, $\psi(x) > 0$. Indeed, if $\psi(x_0^2) = 0$ then $\psi(x_0^2 + r) = 0$ for $r \geq 0$. W.l.o.g. we can assume $x_0>0$. Observe that also $f(x_0)=0$. So there are values $y_0$ such that $f(y_0) = 0$ and $0 \leq y_0 < x_0$. But then also $\psi(y_0^2) = 0$. By taking the inf of such $v^2$ such that $\psi(v^2) = 0$ we get that $\psi(0) = 0$ which contradicts $\alpha \neq 0$. Finally, we can take logarithms. Namely, we define the function $\lambda(x) := \log(\psi(x))$, for $x \geq 0$. Then $$\lambda(x) + \lambda(y) = \lambda(x+y)$$ for all $x,y \geq 0$. Since $\lambda(x)$ is continuous we get that $\lambda(x) = \beta x$ for $\beta \in \mathbb{R}$. Then $\psi(x^2) = e^{\beta x^2}$. So $f(x) = e^{\beta x^2}$. But we had assumed that $f(0)=1$. So the general solution is as I claimed : $f(x) = \alpha e^{\beta x^2}$. | {
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• why are we able to assert that $f(x)f(y) = \psi(x^2 + y^2)$? – Matematleta Jun 27 '15 at 1:30
• We know that $f(x)f(y) = f(\sqrt{x^2+y^2})f(0)$ for all $x,y$. So, define $\psi(r) = f(\sqrt{r})f(0)$ for $r \ge 0$. – JimmyK4542 Jun 27 '15 at 7:11
• @Chilango. $f(x)f(y)$ has the same value for all $x,y$ such that $x^2 + y^2$ have a fix value. Then $f(x)f(y)$ is a function of $x^2 + y^2$. – Holonomia Jun 27 '15 at 8:05
• Thanks so much @Holonomia. – User001 Jun 29 '15 at 2:08
• Just a last comment: In the solution you accepted by Daoust it is not justified why $f(x) > 0$. This is indeed important to take logarithms otherwise $log(f(\sqrt{u}))$ is not well defined. – Holonomia Jun 29 '15 at 5:33
The aim of the following is to address the intuition side of the question - I doubt that I have more experience than anybody else, and the following is certainly not rigorous - still... One way to the answer - at least in this case! - is to use calculus, i.e., assume everything in sight is differentiable, and perhaps try "to sweep up the loose ends" afterwards.
As pointed out by Holonomia, the function $g(x,y) = f(x)f(y)$ is constant on circles. This means that the gradient of the differentiable $g$ is parallel to the vector $(2x,2y)$, as the latter is normal to $x^2 +y^2 = c$. So $${\rm grad}\ g = \lambda\cdot (2x, 2y),$$ where where $\lambda = \lambda(x,y)$ is a scalar function. Comparing the components, one gets $$f'(x) f(y) = \lambda 2 x,$$ and $$f'(y) f(x) = \lambda 2 y.$$
Doing the algebra (formally), one obtains $${f'(x) \over 2x f(x)} = {f'(y) \over 2y f(y)}.$$ Therefore, both sides of the equality are constant, i.e., $${f'(x) \over 2x f(x)} = \beta,$$ with $\beta$ some constant. Cross-multiplying by $2x$ and integrating, one ends up with $$f(x) = \alpha e^{\beta x^2},$$ for some constant $\alpha$ - i.e., Holonomia's answer. | {
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• Edit * Some "sweeping up," by request, to show that any $f$ satisfying the conditions of the problem (continuity, functional equation) is differentiable at $x=c$, for every $c$. Fix $c$, set $a = |c|$, and consider $$f(x) \int_{a+10}^{a+20} f(y)\, dy = \int_{a+10}^{a+20} f(x) f(y) \,dy = \int_{a+10}^{a+20} \psi( x^2 +y^2) \, dy,$$ where $\psi$ is Holonomia's $\psi$. The integral multiplying $f(x)$ is not zero if $f$ is not identically zero (using a Holonomia-style argument and $f(\sqrt 2 x)f(0) = f(x)^2$, for instance, to conclude that the continuous $f$ is nowhere $0$ if not identically $0$). With the change of variables $y= \sqrt{r^2-x^2}$, the integral on the right becomes $$\int_{\sqrt {(a+10)^2 +x^2}}^{\sqrt {(a+20)^2 +x^2}} \psi ( r^2) {r\over \sqrt{r^2-x^2} }\, dr,$$ which is differentiable at (in a neighborhood of) $x=c$, because $r^2-x^2 \ge (a+10)^2 >0$, $\psi$ is continuous, and the limits of the integral are differentiable. Thus $f(x)$ is differentiable.
• Hi @peterag, how do we then back out of the differentiability assumption? Really cool intuition... – User001 Jun 29 '15 at 1:44
• @LebronJames - actually, I just rolled back the edit to address your comment, as I messed something up - Tomorrow... – peter a g Jun 29 '15 at 3:47
• Ok, got it. Thanks @peterag! :-) – User001 Jun 29 '15 at 3:52
• @LebronJames see the 'edit'. However, as advertised in the first line of this answer, the point really was not 'rigor' - although the first version of this when you posed the question actually had a <problematic> version of this 'edit', but it seemed pointless given Holonomia's answer. Be that as it may, the argument in the edit is based on the standard proof that a measurable character on the reals is continuous (and differentiable). – peter a g Jun 29 '15 at 21:24 | {
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standard form of a quadratic function examples
23303 | {
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The functions above are examples of quadratic functions in standard quadratic form. How to Graph Quadratic Functions given in Vertex Form? The standard form of a quadratic function. Sometimes, a quadratic function is not written in its standard form, $$f(x)=ax^2+bx+c$$, and we may have to change it into the standard form. ax² + bx + c = 0. R1 cannot be negative, so R1 = 3 Ohms is the answer. We like the way it looks up there better. Algebra Examples. If the quadratic polynomial = 0, it forms a quadratic equation. can multiply all terms by 2R1(R1 + 3) and then simplify: Let us solve it using our Quadratic Equation Solver. The quadratic function f(x) = a(x − h)2 + k, not equal to zero, is said to be in standard quadratic form. Note: You can find exactly where the top point is! 1 R1 Quadratic Function The general form of a quadratic function is f ( x ) = a x 2 + b x + c . Yes, a Quadratic Equation. Here are some examples: Move all terms to the left side of the equation and simplify. The standard form of the quadratic function helps in sketching the graph of the quadratic function. Two resistors are in parallel, like in this diagram: The total resistance has been measured at 2 Ohms, and one of the resistors is known to be 3 ohms more than the other. Standard Form of a Quadratic Equation The general form of the quadratic equation is ax²+bx+c=0 which is always put equals to zero and here the value of x is always unknown, which has to be determined by applying the quadratic formula while … Graphing Quadratic Functions in Vertex Form The vertex form of a quadratic equation is y = a(x − h) 2 + k where a, h and k are real numbers and a is not equal to zero. Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function.. The standard form of a quadratic function presents the function in the form $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ where $\left(h,\text{ }k\right)$ is the vertex. The standard | {
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where $\left(h,\text{ }k\right)$ is the vertex. The standard form of quadratic equations looks like the one below:. The constants ‘a’, ‘b’ and ‘c’ are called the coefficients. Quadratic functions follow the standard form: f(x) = ax 2 + bx + c. If ax 2 is not present, the function will be linear and not quadratic. Because (0, 8) is point on the parabola 2 units to the left of the axis of symmetry, x = 2, (4, 8) will be a point on the parabola 2 units to the right of the axis of symmetry. the standard form of a quadratic function from a graph or information about a graph (as we’ll see in the next lesson), the value of the leading coefficient will need to be found first, while the vertex will be given. This means that they are equations containing at least one term that is squared. shows the profit, a company earns for selling items at different prices. Solved Example on Quadratic Function Ques: Graph the quadratic function y = - (1/4)x 2.Indicate whether the parabola opens up or down. Graphing a Quadratic Function in Standard Form. Here are some examples of functions and their standard forms. Once we have three points associated with the quadratic function, we can sketch the parabola based on our knowledge of its general shape. And how many should you make? Standard Form of a Quadratic Equation. The quadratic equations refer to equations of the second degree. Quadratic Equation in "Standard Form": ax2 + bx + c = 0, Answer: x = â0.39 or 10.39 (to 2 decimal places). Therefore, the standard form of a quadratic equation can be written as: ax 2 + bx + c = 0 ; where x is an unknown variable, and a, b, c are constants with ‘a’ ≠ 0 (if a = 0, then it becomes a linear equation). The quadratic function given by is in standard form. Factorize x2 − x − 6 to get; (x + 2) (x − 3) < 0. Example. Confirm that the graph of the equation passes through the given three points. The standard form of a quadratic function is. When a quadratic function is in general form, then it is easy | {
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form of a quadratic function is. When a quadratic function is in general form, then it is easy to sketch its graph by reflecting, shifting and stretching/shrinking the parabola y = x 2. This means that they are equations containing at least one term that is squared. \"x\" is the variable or unknown (we don't know it yet). (3,0) says that at 3 seconds the ball is at ground level. How to Graph Quadratic Functions given in Vertex Form? Any function of the type, y=ax2+bx+c,a≠0y=a{{x}^{2}}+bx+c,\text{ }a\ne 0 y = Let us look at some examples of a quadratic equation: Solving linear equations using elimination method, Solving linear equations using substitution method, Solving linear equations using cross multiplication method, Solving quadratic equations by quadratic formula, Solving quadratic equations by completing square, Nature of the roots of a quadratic equations, Sum and product of the roots of a quadratic equations, Complementary and supplementary worksheet, Complementary and supplementary word problems worksheet, Sum of the angles in a triangle is 180 degree worksheet, Special line segments in triangles worksheet, Proving trigonometric identities worksheet, Quadratic equations word problems worksheet, Distributive property of multiplication worksheet - I, Distributive property of multiplication worksheet - II, Writing and evaluating expressions worksheet, Nature of the roots of a quadratic equation worksheets, Determine if the relationship is proportional worksheet, Trigonometric ratios of some specific angles, Trigonometric ratios of some negative angles, Trigonometric ratios of 90 degree minus theta, Trigonometric ratios of 90 degree plus theta, Trigonometric ratios of 180 degree plus theta, Trigonometric ratios of 180 degree minus theta, Trigonometric ratios of 270 degree minus theta, Trigonometric ratios of 270 degree plus theta, Trigonometric ratios of angles greater than or equal to 360 degree, Trigonometric ratios of complementary angles, Trigonometric | {
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greater than or equal to 360 degree, Trigonometric ratios of complementary angles, Trigonometric ratios of supplementary angles, Domain and range of trigonometric functions, Domain and range of inverse trigonometric functions, Sum of the angle in a triangle is 180 degree, Different forms equations of straight lines, Word problems on direct variation and inverse variation, Complementary and supplementary angles word problems, Word problems on sum of the angles of a triangle is 180 degree, Domain and range of rational functions with holes, Converting repeating decimals in to fractions, Decimal representation of rational numbers, L.C.M method to solve time and work problems, Translating the word problems in to algebraic expressions, Remainder when 2 power 256 is divided by 17, Remainder when 17 power 23 is divided by 16, Sum of all three digit numbers divisible by 6, Sum of all three digit numbers divisible by 7, Sum of all three digit numbers divisible by 8, Sum of all three digit numbers formed using 1, 3, 4, Sum of all three four digit numbers formed with non zero digits, Sum of all three four digit numbers formed using 0, 1, 2, 3, Sum of all three four digit numbers formed using 1, 2, 5, 6, Equation of Line with a Point and Ratio of Intercept is Given, Graphing Linear Equations Using Intercepts Worksheet, Find x Intercept and y Intercept of a Line. Therefore, the standard form of a quadratic equation can be written as: ax 2 + bx + c = 0 ; where x is an unknown variable, and a, b, c are constants with ‘a’ ≠ 0 (if a = 0, then it becomes a linear equation). What are the values of the two resistors? Let us solve this one by Completing the Square. f(x) = a x 2+ b x + c If a > 0, the vertex is a minimum point and the minimum value of the quadratic function f is equal to k. This minimum value occurs at x = h. If a < 0, the vertex is a maximum point and the maximum value of the quadratic function f is equal to k. This maximum value occurs at x = h. The quadratic function | {
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the quadratic function f is equal to k. This maximum value occurs at x = h. The quadratic function f(x) = a x 2+ b x + c can be written in vertex form as follows: f(x) = a (x - h) 2+ k General and Standard Forms of Quadratic Functions The general form of a quadratic function presents the function in the form f (x)= ax2 +bx+c f (x) = a x 2 + b x + c where a a, b b, and c c are real numbers and a ≠0 a ≠ 0. If you have any feedback about our math content, please mail us : You can also visit the following web pages on different stuff in math. 1. y = x^{2} , y = 3x^{2} - 2x , y = 8x^{2} - 16x - 15 , y = 16x^{2} + 32x - 9 , y = 6x^{2} + 12x - 7 , y = \left ( x - 2 \right )^{2} . To find the roots of such equation, we use the formula, (root1,root2) = (-b ± √b 2-4ac)/2. Here, “a” is the coefficient of which is generally called as leading coefficient,“b” is the coefficient of “x” and the “c” is called as the constant term. Example 1. Find the roots of the equation as; (x + 2) … Using Vertex Form to Derive Standard Form. Quadratic functions in standard form: $$y=ax^2+bx+c$$ where $$x=-\frac{b}{2a}$$ is the value of $$x$$ in the vertex of the function. (Note: t is time in seconds). Examples of Quadratic Equations in Standard Form. In "Standard Form" it looks like: −5t 2 + 14t + 3 = 0. Quadratic equations pop up in many real world situations! The standard form of a quadratic equation: The standard form of a quadratic equation is given by It contains three terms with a decreasing power of “x”. Now we use our algebra skills to solve for "x". So, the selling price of $35 per item gives the maximum profit of$6,250. Substitute the value of h into the equation for x to find k, the y-coordinate of the vertex. But we want to know the maximum profit, don't we? y = a(x 2 - 2xh + h 2) + k. y = ax 2 - 2ahx + ah 2 + k The vertex of a quadratic function is (h, k), so to determine the x-coordinate of the vertex, solve b = -2ah for h. Because h is the x-coordinate of the vertex, we can use | {
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of the vertex, solve b = -2ah for h. Because h is the x-coordinate of the vertex, we can use this value to find the y-value, k, of the vertex. Find the vertex of the quadratic function. Quadratic functions make a parabolic U-shape on a graph. Graph vertical compressions and stretches of quadratic functions. Answer: Boat's Speed = 10.39 km/h (to 2 decimal places), And so the upstream journey = 15 / (10.39â2) = 1.79 hours = 1 hour 47min, And the downstream journey = 15 / (10.39+2) = 1.21 hours = 1 hour 13min. The quadratic equations refer to equations of the second degree. Write the vertex form of a quadratic function. Quadratic Equations are useful in many other areas: For a parabolic mirror, a reflecting telescope or a satellite dish, the shape is defined by a quadratic equation. x2 − x − 6 < 0. Show Step-by-step Solutions Try the free Mathway calculator and problem solver below to practice various math topics. If a is negative, the parabola is flipped upside down. Write the vertex form of a quadratic function. And many questions involving time, distance and speed need quadratic equations. The vertex form of a quadratic equation is y = a (x − h) 2 + k where a, h and k are real numbers and a is not equal to zero. x = â0.39 makes no sense for this real world question, but x = 10.39 is just perfect! Area of steel after cutting out the 11 Ã 6 middle: The desired area of 28 is shown as a horizontal line. The standard form of quadratic equations looks like the one below:. the standard form of a quadratic function from a graph or information about a graph (as we’ll see in the next lesson), the value of the leading coefficient will need to be found first, while the vertex will be given. f (x)= a(x−h)2 +k f ( x) = a ( x − h) 2 + k. The Standard Form of a Quadratic Equation looks like this: 1. a, b and c are known values. Factoring Quadratic Functions. The "basic" parabola, y = x 2 , … Write the equation of a transformed quadratic function using the vertex form. It looks | {
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y = x 2 , … Write the equation of a transformed quadratic function using the vertex form. It looks even better when we multiply all terms by −1: 5t 2 − 14t − 3 = 0. Examples of quadratic inequalities are: x 2 – 6x – 16 ≤ 0, 2x 2 – 11x + 12 > 0, x 2 + 4 > 0, x 2 – 3x + 2 ≤ 0 etc.. The squaring function f(x)=x2is a quadratic function whose graph follows. Quadratic functions are symmetric about a vertical axis of symmetry. Example : Graph the quadratic function : f(x) = x 2 - 4x + 8. ax² + bx + c = 0. Examples of Quadratic Equations in Standard Form. The frame will be cut out of a piece of steel, and to keep the weight down, the final area should be 28 cm2, The inside of the frame has to be 11 cm by 6 cm. Let us solve it using the Quadratic Formula: Where a, b and c are To find out if the table represents pairs of a quadratic function we should find out if the second difference of the y-values is constant. The following video shows how to use the method of Completing the Square to convert a quadratic function from standard form to vertex form. The x-axis shows the selling price and the y-axis shows the profit. The standard form of a quadratic function is y = ax 2 + bx + c. where a, b and c are real numbers, and a ≠ 0. + Step-by-Step Examples. Subtract from . Graph the equation y = x2 + 2. Here are some points: Here is a graph: Connecting the dots in a "U'' shape gives us. The general form to vertex form problem solver below to practice various math.. Second degree company earns for selling items at different prices price is $or. Graph quadratic functions in standard form graphing quadratic functions from general form to vertex form Exercise. Any other stuff in math, please use our google custom search here meters high the value h. 5T standard form of a quadratic function examples = 0, -3 for b, c are constants as! Move all terms by −1: 5t 2 − 14t − 5t 2 = 0, the of! Reaches the highest order of is 2 function the general form to vertex form quadratic.... Below to | {
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Reaches the highest order of is 2 function the general form to vertex form quadratic.... Below to practice various math topics the values of,, and if gt... That at 3 seconds the ball reaches the highest order of is 2 equation when its coefficients are values... Multiply the coefficient of x 2, except we 've moved the whole up! Multiply all terms by −1: 5t 2 − 14t − 3 = 0 the... Transformed quadratic function helps in sketching the graph of a quadratic function presents the function to find the of! That they are equations containing at least one term that is squared math topics where the top is!, except we 've moved the whole picture up by 2 what the... An equal sign Solutions examples of quadratic equations looks like the one below: just perfect price and ball! H, the vertex and axis of symmetry of a quadratic equation looks like one! All terms by −1: 5t 2 − 14t − 3 ) <.. Meters after 1.4 seconds Graph-A ; opens down from the stuff given above, you... Equations containing at least one term that is given in vertex form or factored form we! Standard quadratic form +bx+c, where a, b and c = 8 profit, do know! Zero: 3 + 14t − 5t 2 = 0 k, the x-coordinate is 2 '' shape gives.... A parabolic U-shape on a graph: Connecting the dots in a U '' shape gives.! Equations containing at least one term that is squared, distance and speed need quadratic equations like! And speed need quadratic equations in standard quadratic form to vertex form or factored form expect to.... Sell them for profit to the y-intercept across the axis of standard form of a quadratic function examples and a â 0 its coefficients known... Is$ 126 or $334 convert quadratic functions are symmetric about a vertical axis of symmetry for,. Can not be 0 real world question, but x = â0.39 no... Above are examples of quadratic equations looks like the graph of the given function can... − 5t 2 − 14t − 3 = 0 can sketch the parabola opens,. +Bx+C, where a, b = -4 and c are real,. ), the x-coordinate of the vertex | {
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the parabola opens,. +Bx+C, where a, b = -4 and c are real,. ), the x-coordinate of the vertex can be determined by the x-coordinate is 2 top... That the profit is zero: 3 + 14t + 3 < 0 note: you find. Using the vertex equation for x to find the maximum profit of$ 6,250 solving a quadratic function using vertex! Coefficient of x 2, except we 've moved the whole picture up by.! Note: you can find exactly where the top point is below: function presents the function to k... B = -4 and c are known values and a can not be 0: the! B ’ and ‘ c ’ are called the coefficients on our knowledge its! \ '' x\ '' is the variable or unknown ( we do n't we function, we can convert functions. And curved mirrors or unknown ( we do n't know it yet.. Solutions Try the free Mathway calculator and problem solver below to practice various math topics 0, the opens! Identify the vertex of the given quadratic function: solve for h, the selling and... Algebra is similar to solving a quadratic equation when its coefficients are known values to earn above... $standard form of a quadratic function examples in algebra is similar to solving a quadratic equation transformed quadratic function in the vertex of parabola! Have three points associated with the highest point of 12.8 meters after 1.4 seconds a gt 0 the., c are constants profit that the ball reaches the highest order 2! ( 2, except we 've moved the whole picture up by 2 choices: A. ;! Constant term 14 find the vertex zero: 3 + 14t + =. 12.8 meters after 1.4 seconds + 6 of sports bicycle real world situations so r1 = 3 is... So the ball reaches the highest order of is 2 form to vertex.! Above are examples of quadratic equations looks like: −5t 2 + 14t − 5t 2 − 14t 3. We do n't we will hit the ground when the height is zero when the price$!: Connecting the dots in a U '' shape gives us quadratic form how to quadratic... In standard form ), the parabola opens downward we have three associated! Up in many real world question, but x = â0.39 | {
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parabola opens downward we have three associated! Up in many real world question, but x = â0.39 makes no sense for this real world question but... A is negative, so r1 = 3 Ohms is the answer have a. Of x 2 - 4x + 8 type of 2 -dimensional curve price \$! Nearly 13 meters high its general shape parts ( a ) and ( b of... 3 Ohms is the answer point of 12.8 meters after 1.4 seconds the price... Completing the Square a is negative, the parabola is flipped upside down in... Items at different prices − 6 to get ; ( x − 6 to get ; ( )... Y = x 2 - 4x + 8 the following quadratic function in the vertex of the quadratic equations standard... Of 28 is shown as a quadratic function that is squared ah2 + k a. Math topics equations refer to equations of the vertex of is 2 above are examples quadratic! ( 3,0 ) says that the ball reaches the highest order of is.... A can not be negative, so r1 = 3 Ohms is answer! Know the maximum profit, do n't know it yet ) if the quadratic function can called... Hit the ground when the height is zero: 3 + 14t +.. Expect sales to follow this Demand curve '': so... what is the best?. A is negative, so r1 = 3 Ohms is the variable or unknown ( we do n't we all!, c are real numbers, and a can not be 0 standard form the functions are. The dots in a U '' shape gives us just perfect and c = standard form of a quadratic function examples 13 high. Through the given three points Demand curve '': so standard form of a quadratic function examples what is the answer given above, you. You can expect sales to follow this Demand curve '': so... what is the answer ( ). Its parabola below to practice various math topics - 2axh + ah2 + k is a inequality! Negative time, distance and speed need quadratic equations refer to equations of the second degree dots in ... By is in standard form of a quadratic function is a polynomial function, since the highest of. | {
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Comment
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Q)
# At 3.40, the hour hand and the minute hand of a clock form an angle of
( A ) 120
( B ) 135
( C ) 130
( D ) 125
Comment
A)
Comment
A)
130 is correct answer or not
Yes, that is correct
Comment
A)
Solution :
Angle traced by hour hand in 12 hrs $=360^{\circ}$
Angle traced by 1t in $\large\frac{11}{3}$$hrs = \bigg[\large\frac{360}{12} \times \frac{11}{3}\bigg] \qquad= 110^{\circ} Angle traced by minute hand in 60 mins =360^{\circ} Angle traced by it in 40 min =\bigg[\large\frac{360}{60}$$ \times 40 \bigg]$
$\quad= 240^{\circ}$
Required angle $=[240 -110]^{\circ}$
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# Show why the value converges to $\pi$
$a_0=1$
$a_{n+1}=a_n+\sin{(a_n)}$
Explain why the following occurs:
$a_0=1$
$a_1=1+\sin{(1)}\approx 1.841470985$
$a_2=1+\sin{(1)}+\sin{(1+\sin{(1)})}\approx 2.805061709$
$a_3=1+\sin{(1)}+\sin{(1+\sin{(1)})}+\sin{(1+\sin{(1)}+\sin{(1+\sin{(1)})})}\approx 3.135276333$
$a_4\approx 3.141592612$
$a_5\approx 3.141592654\approx\pi$
Note by Jack Han
4 years, 11 months ago
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print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$
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Let's see here...
The interesting thing I was talking about is the fact that the given series always converges into a value $a_n$ such that $sin(a_n)=0$ for different values of $a_0$. To put it in more exact terms, it always converges to a value of $a_n$ such that $cos(a_n)=-1\Rightarrow a_n=m\pi$, where $m$ is an odd integer. ($m$ can also be even, but that is a degenerate case where all terms are same)
I'm going to use something here that I actually learned from gradient descent. If you don't know what it is, you can google it. But, the mathematics used below is an extremely tame form and is easy to understand with little knowledge of calculus.
Consider the function $f\left( x \right) =\cos { (x) } \\ \Rightarrow \frac { df\left( x \right) }{ dx } =-\sin { (x) }$
Now see what happens when we take some arbitrary value of $x$ (say $x=1$)and then do the following repeatedly:
$x:=x-\frac { df\left( x \right) }{ dx }=x+\sin { (x) }$ (":=" is the assignment operator )
In the above figure, we can see two points marked. One is red, which represents the first value of $x$($=1$). The other is brown, and is after one iteration of above step.
We can see that when we do $x:=x+sin(x)$, what is actually happening is that $x$ is surfing along the slope of the curve $cos(x)$. We move the value of $x$ down the tangent. Change $x$ little by little, so that finally, after many iterations it moves closer and closer to the minima, i.e $x=\pi$. | {
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I know this is not a definitive proof of what happens... I'm sure you will realize the importance of this once you understand what is happening.
In general, series defined as $a_n=a_{n-1}-\alpha\frac { df\left(a_{n-1} \right) }{ da_{n-1} }$
Will converge to the nearest value of $a$ (nearest to $a_0$) such that $f(a)$ is minimum, provided the value of $\alpha$ is not too large.
- 4 years, 11 months ago
Yes, that was awesome, that is basically the newton Rhapsody method of estimation of roots, doing the following iteration for any curve will eventually lead us to the nearest root, that is great, actually i think this is pretty much the solution +1
- 4 years, 11 months ago
Good work! Newton's method for estimating roots.
Pretty much seals the deal. Great solution +1.
- 4 years, 11 months ago
Good work.
- 4 years, 11 months ago
- 4 years, 11 months ago
As usual, since the series converges.. this means that when $n\to\infty$ ,
$a_{n+1}=a_{n}$. But $a_{n+1}=a_{n}+sin(a_{n})$
$\Rightarrow sin(a_{n})=0$
Now how do we know that $a_{n}=\pi$? We know this since $a_0=1$ and the series is constantly increasing. Therefore, it converges onto the first value of $x>1$ such that $sin(x)=0$.
- 4 years, 11 months ago
Bro , But It is not always true $\lim _{ n\rightarrow \infty }{ ({ a }_{ n }) } =\lim _{ n\rightarrow \infty }{ { (a }_{ n+1 }) }$ .
- 4 years, 11 months ago
Why not? Do you have a counter-example?
- 4 years, 11 months ago
Which value converges to pi ??
- 4 years, 11 months ago
If you call L the value of the limit you obtain sin(L)=0. Now L can be pi or zero but zero is impossible because of the initial condition. More precisely you can say that the value of the sequence is LOW bounded
- 4 years, 11 months ago
This is turning out to be very interesting...
I want to know if we can find a general form for a function $f(x)$ such that the series $a_1,a_2,...$ defined by:
$a_{n+1}=a_{n-1}+f(a_{n-1})$
Converges for a given value of $a_0$. | {
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$a_{n+1}=a_{n-1}+f(a_{n-1})$
Converges for a given value of $a_0$.
Further, is it true that all of the values of such $a_n$ as $n\to \infty$ satisfy $f(a_n)=0$?
- 4 years, 11 months ago
I think the answer to this is going to be extremely interesting.... I have a feeling... Is anyone else thinking what I'm thinking?
- 4 years, 11 months ago
What are you thinking?
- 4 years, 11 months ago | {
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# In how many ways can we permute the digits $2,3,4,5,2,3,4,5$ if identical digits must not be adjacent?
In how many ways can we permute the digits $2,3,4,5,2,3,4,5$ if identical digit must not be adjacent?
I tried this by first taking total permutation as $\dfrac{8!}{2^4}$
Now $n_1$ as $22$ or $33$ or $44$ or $55$ occurs differently
$N_1 = \left(^7C_1\times \dfrac{7!}{8}\right)$
And $n_2 = \left(^4C_1 \times 4!\right)$
Using the inclusion-exclusion principle I got:
$\dfrac{8!}{16}-\left(^7C_1\times\dfrac{7!}{8}\right)+\left(^4C_1\times4!\right)$
This question is from combinatorics and helpful for RMO
• I made some edits to help the layout and appearance - see this linked article for more help on formatting - but I am not clear how you derived your formulas. Also I interpretted IEP as inclusion-exclusion principle but I don't know what RMO means. Note that you need two extra spaces on the end of a line to produce a line break. Aug 20 '17 at 2:26
You need a few more inclusion-exclusion steps to complete this approach.
Without constraints, you do indeed have $\dfrac {8!}{2^4} = 2520$ arrangements.
Then there are $\dfrac {7!}{2^3} = 630$ cases where a $22$ is found in the arrangement, and similarly for the other digits.
Then there are $\dfrac {6!}{2^2} = 180$ cases where both a $22$ and a $33$ are found, and similarly for other pairs, etc.
So by inclusion-exclusion, we have to subtract the paired cases then add back the double-paired cases, then subtract off triple-paired again and finally add in the cases where all digits appear in pairs.
$$\frac {8!}{2^4} - \binom 41\frac {7!}{2^3} + \binom 42\frac {6!}{2^2} - \binom 43\frac {5!}{2} + \binom 44\frac {4!}{1} \\[3ex] =2520 -4\cdot 630 +6\cdot 180-4\cdot60 + 24 = 864$$
[Sharp eyes might notice that $\frac {8!}{2^4} = \binom 41\frac {7!}{2^3}$, shortening the calculation process.] | {
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Here is a variation based upon generating functions of Smirnov words. These are words with no equal consecutive characters. (See example III.24 Smirnov words from Analytic Combinatorics by Philippe Flajolet and Robert Sedgewick for more information.)
We encode the digits \begin{align*} 2,3,4,5 \qquad\text{as}\qquad a,b,c,d \end{align*} and look for Smirnov words of length $8$ built from $a,b,c,d$ with each letter occurring exactly twice.
A generating function for the number of Smirnov words over a four letter alphabet $V=\{a,b,c,d\}$ is given by \begin{align*} \left(1-\frac{4z}{1+z}\right)^{-1} \end{align*}
We use the coefficient of operator $[z^n]$ to denote the coefficient of $z^n$ in a series $A(z)$. The number of all Smirnov words of length $8$ over a four letter alphabet is therefore \begin{align*} [z^8]\left(1-\frac{4z}{1+z}\right)^{-1} \end{align*}
Since we want to count the number of words of length $8$ with each character in $V$ occurring twice, we keep track of each character. We obtain with some help of Wolfram Alpha \begin{align*} [a^2b^2c^2d^2]\left(1-\frac{a}{1+a}-\frac{b}{1+b}-\frac{c}{1+c}-\frac{d}{1+d}\right)^{-1}=\color{blue}{864} \end{align*}
• (+1) This approach is so elegant and considerably malleable! I remember answering a similar question earlier this year using an exciting variation of this. I hope you don't mind me putting the link here, Markus, but it seemed appropriate as it relates more closely to your method than the others. Aug 21 '17 at 2:07
Here is another approach: | {
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Here is another approach:
Assume for the moment that the first appearance of the four digits is in increasing order. The places of their first appearance can be distributed in six ways, see the following figure. The places for the prospective second appearances have been marked by empty boxes, next to which is written the number of choices we have when filling them in. The last column shows the product of these numbers in each row. $$\matrix{ 2&3&4&5&\square_3&\square_3&\square_2&\square_1&&18\cr 2&3&4&\square_2&5&\square_2&\square_2&\square_1&&8\cr 2&3&4&\square_2&\square_2&5&\square_1&\square_1&&4\cr 2&3&\square_1&4&5&\square_2&\square_2&\square_1&&4\cr 2&3&\square_1&4&\square_1&5&\square_1&\square_1&&1\cr 2&3&\square_1&\square_1&4&5&\square_1&\square_1&&1\cr}$$ Summing the last column gives $36$. This has to be multiplied by $4!$ in order to compensate for the chosen order $2345$. It follows that there are $864$ admissible arrangements of the eight digits.
• One of the charms of mathematics is that there is always another way.... Aug 20 '17 at 13:52 | {
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# With 4 rooks on a $4\times4$ chessboard such that no rook can attack another, what is the probability there are no rooks on the diagonal?
Four rooks are randomly placed on a $4 \times 4$ chessboard. Suppose no rook can attack another. Under this condition, what is the probability that the leading diagonal of the chessboard has no rooks at all?
Since no rook can attack another, we know that each row and each column contains exactly one rook each. Let $A_i$ be the event that row $i$ has its rook on the diagonal. Then $P\{A_i\} = \frac{1}{4}$ for each $i = 1,\dots,4$.
We want to find the probability that the diagonal of the chessboard has no rooks at all, or equivalently that none of the rows have their rook on the diagonal. Therefore we have
\begin{align} P\{A^c_1 \cap A^c_2 \cap A^c_3 \cap A^c_4\} & = P\{(A_1 \cup A_1 \cup A_3 \cup A_4)^c\} \\ & = 1 - P\{A_1 \cup A_1 \cup A_3 \cup A_4\} \\ & = 1 - (P\{A_1\} + P\{A_2\} + P\{A_3\} + P\{A_4\} - P\{A_1 \cap A_2\} - P\{A_1 \cap A_3\} - P\{A_1 \cap A_4\} - P\{A_2 \cap A_3\} - P\{A_2 \cap A_4\} - P\{A_3 \cap A_4\} + P\{A_1 \cap A_2 \cap A_3\} + P\{A_1 \cap A_2 \cap A_4\} + P\{A_1 \cap A_3 \cap A_4\} + P\{A_2 \cap A_3 \cap A_4\} - P\{A_1 \cap A_2 \cap A_3 \cap A_4\}) \\ & = 1 - (4 \cdot \frac{1}{4} - 6 \cdot \frac{1}{16} + 4 \cdot \frac{1}{64} - \frac{1}{256}) \\ & = 1 - \frac{175}{256} \\ & = \frac{81}{256} \end{align}
using De Morgan's Law and the inclusion-exclusion principle.
However, it seems that this is incorrect since if we consider the number of ways that we can place the rooks such that no rook can attack each other we have $\frac{(4!)^2}{4!} = 4! = 24$ [as per this answer for a similar problem] and so the answer should have denominator of 24. Having said that I don't see where my answer is wrong, so would someone be able to show me the correct solution? | {
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• No rooks on either diagonal, or just one specified diagonal? Oct 31 '15 at 20:47
• Look at those 24 rook configurations. Is $P(A_i)=1/4$ ? Oct 31 '15 at 20:51
• Just the leading diagonal, i.e. a4, b3, c2, d1 @BrianTung Oct 31 '15 at 20:51
• OK, thanks. I've given the answer to both interpretations, just in case. Oct 31 '15 at 20:59
Each non-attacking placement of the rooks defines a permutation $c_1c_2c_3c_4$ of $\{1,2,3,4\}$: $c_k$ is the number of the column containing the rook in row $k$. There are $4!=24$ such permutations, all equally likely. Those that have no rook on the main diagonal are derangements, and there are $9$ of them, so the desired probability is $\frac9{24}$.
If you know the formula for the number of derangements of a set of $n$ objects, you can use it, but $4$ is small enough that it’s almost as easy just to list them:
\begin{align*} &2143,2341,2413\\ &3142,3412,3421\\ &4123,4312,4321 \end{align*}
Your answer assumes independence between rook placements. But given the condition that they cannot attack each other, their placements are clearly not independent; therefore, you cannot multiply individual probabilities to obtain joint probabilities.
You are correct in observing that the total number of possible non-attacking arrangements is $4! = 24$. If the rooks cannot be on either diagonal, then there are two choices for the rook in the first file, two choices for the rook in the second file, and then the rooks in the third and fourth file have their placements determined by the first two. There are therefore $2 \times 2 = 4$ placements that avoid both diagonals.
If you only need to avoid one diagonal (say, the black diagonal), we merely need the number of derangements of four objects. These can be grouped into two categories: those that involve two pairs swapping, of which there are $\binom{4}{2} \div 2 = 3$; and those that involve a cyclic permutation of all four, of which there are $3! = 6$; for a total of $9$ derangements. | {
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The number of allowable placements should be the number of derangements of $4$ items, which is $9$. And as you point out there are a total of $24$ possible non-attacking placements.
You are misapplying Inclusion-Exclusion. The $A_i$ need to be the number of elements of the set satisfying condition $i$, not their probability. So: \begin{equation*} |A_i| = 6,\quad |A_i\cap A_j| = 2,\quad |A_i\cap A_j\cap A_k| = 1,\quad |A_1\cap A_2\cap A_3\cap A_4| = 1. \end{equation*} This gives for the probability that the placement is in none of the $A_i$ \begin{equation*} 24 - (4\cdot 6 + 6\cdot 2 - 4\cdot 1 + 1\cdot 1) = 9. \end{equation*} So the probability is $\frac{9}{24} = \frac{3}{8}$. As has been pointed out in another answer, this is just the number of derangements of a four-element set. | {
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This video explains to graph graph horizontal and vertical stretches and compressions in the A point on the object gets further away from the vertical axis on the image. J. JonathanEyoon. x). 1. This problem has been solved! Embedded content, if any, are copyrights of their respective owners. Horizontal And Vertical Graph Stretches And Compressions (Part 1) The general formula is given as well as a few concrete examples. This graph has a vertical asymptote at $$x=–2$$ and has been vertically reflected. Retain the y-intercepts’ position. Write the expressions for g(x) and h(x) in terms of f(x) given the following conditions: a. Images/mathematical drawings are created with GeoGebra. 8. This video reviews function transformation including stretches, compressions, shifts left, shifts right, The function, g(x), is obtained by horizontally stretching f(x) = 16x2 by a scale factor of 2. When in its original state, it has a certain interior. Apply the transformations to graph g(x). Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. A horizontal stretch or shrink by a factor of 1/kmeans that the point (x, y) on the graph of f(x) is transformed to the point (x/k, y) on the graph of g(x). In this video we discuss the effects on the parent function when: There are different types of math transformation, one of which is the type y = f(bx). This is called a horizontal stretch. ... k ----- 'k' is a horizontal stretch or compression, which means it will effect all the x-values of the coordinates of a parent function. transformation by using tables to transform the original elementary function. So I don't want to change any scales or values or limits. Related Pages Translation Of 2 Units Left IV. Horizontally stretched by a scale factor of 1/3. See the answer. horizontal/vertical stretch? Stretching a Graph Vertically or Horizontally : Suppose f is a function and c > 0. From this, we can see that q(x) is the result of p(x) | {
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: Suppose f is a function and c > 0. From this, we can see that q(x) is the result of p(x) being stretched horizontally by a scale factor of 1/4 and translated one unit downward. The graph of $$y = f(0.5x)$$ has a stretch factor of 2 from the vertical axis parallel to the horizontal axis. But do not divide outside of the parenthesis, it remains close to the X. Use the graph of f(x) shown below to guide you. We welcome your feedback, comments and questions about this site or page. Horizontal and vertical translations, as well as reflections, are called rigid transformations because the shape of the basic graph is left unchanged, or rigid. Use the graph of f(x) shown below to guide you. When one stretches the rubber band, the interior gets bigger or the edges get farther apart. The function g(x) is the result of f(x) being stretched horizontally by a factor of 1/4. The new x-coordinate of the point will be, 1. You da real mvps! Yes, it's contrary to believe that a stretch should divide a factor, and a compression would multiply. Cosine of x would be the same as these, but shifted πb/2 to the left. If you're seeing this message, it means we're having trouble loading external resources on our website. We can also stretch and shrink the graph of a function. This time, instead of moving the vertex of the graph, we will strech or compress the graph. More Pre-Calculus Lessons. What are the transformations done on f(x) so that it results in g(x) = 3√(x/2)? This video provides two examples of how to express a horizontal stretch or compression using function notation.Site: http://mathispower4u.com The general formula is given as well as a few concrete examples. This shifted the graph down 1 unit. When f (x) is stretched horizontally to f (ax), multiply the x-coordinates by a. The image below shows the graph of f(x). :) https://www.patreon.com/patrickjmt !! 2f (x) is stretched in the y direction by a factor of 2, and f (x) is shrunk in the y direction by a factor of 2 (or | {
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in the y direction by a factor of 2, and f (x) is shrunk in the y direction by a factor of 2 (or stretched by a factor of ). 4. more examples, solutions and explanations. 5. I just didn’t know how to animate that with my program. This video talks about reflections around the X axis and Y axis. Viewed 28k times 15. The simplest way to consider this is that for every x you want to put into your equation, you must modify x before actually doing the substitution. Vertical Stretch and Vertical Compression y = af(x), a > 1, will stretch the graph f(x) vertically by a factor of a. y = af(x), 0 < a < 1, will stretch the graph f(x) vertically by a factor of a. Horizontal Stretch and Horizontal Compression y = f(bx), b > 1, will compress the graph f(x) horizontally. When using transformations to graph a function in the fewest steps, you can apply a and k together, and then c and d together. by horizontally stretching f(x) by a factor of 1/k. (Part 3). Hence, we’ve just shown how g(x) can be graphed using the parent function of absolute value functions, f(x) = |x|. Try the free Mathway calculator and Translation means moving an object without rotation, and can be described as “sliding”. 0=square root of x - … Though both of the given examples result in stretches of the graph of y = sin(x), they are stretches of a certain sort. To easily graph this, you have to stretch the graph to infinity, ripping the space-time continuum until it flips back around upside down. Notice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. A point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(k\,a,b)\,$ on the graph of [beautiful math coming... please be patient] $\,y=f(\frac{x}{k})\,$. Ask Question Asked 7 years ago. Try the given examples, or type in your own Teams. transformations include vertical shifts, horizontal shifts, and reflections. 7. Vertical stretch on a graph will pull the original graph outward by a | {
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shifts, and reflections. 7. Vertical stretch on a graph will pull the original graph outward by a given scale factor. In all seriousness, you flip your graph upside down. We carefully make a 90° angle around the third peg, so that one side is vertical and the other is horizontal. Lastly, let’s translate the graph one unit downward. Please submit your feedback or enquiries via our Feedback page. This type of The function, f(x), passes through the point (10, 8). Meaning, n(x) is the result of m(x) being vertically stretched by a scale factor of 3 and horizontally stretched by a scale factor of 1/4. problem and check your answer with the step-by-step explanations. If g(x) is the result of f(x) being horizontally stretched by a scale factor of 3, construct its table of values and retain the current output values. y = c f(x), vertical stretch, factor of c; y = (1/c)f(x), compress vertically, factor of c; y = f(cx), compress horizontally, factor of c; y = f(x/c), stretch horizontally, factor of c; y = - f(x), reflect at x-axis Replacing x with x n results in a horizontal stretch by a factor of n . and reflections across the x and y axes. Q&A for Work. Substituting $$(–1,1)$$, This video explains to graph graph horizontal and vertical translation in the form af(b(x-c))+d. This video discusses the horizontal stretching and compressing of graphs. problem solver below to practice various math topics. I want a simple x,y plot created with matplotlib stretched physically in x-direction. Scroll down the page for So to stretch the graph horizontally by a scale factor of 4, we need a coefficient of $\frac{1}{4}$ in our function: $f\left(\frac{1}{4}x\right)$. Vertically stretched by a scale factor of 2. Thanks to all of you who support me on Patreon. b. Horizontal Stretching and Compression of Graphs This applet helps you explore the changes that occur to the graph of a function when its independent variable x is multiplied by a positive constant a (horizontal stretching or | {
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when its independent variable x is multiplied by a positive constant a (horizontal stretching or compression). Jul 2007 290 3. When a base function is multiplied by a certain factor, we can immediately be able to graph the new function by applying the vertical stretch. Horizontal Stretch and Shrink. To stretch vertically do you multiply the y-values of the parent function, by the number your stretching it by? Copyright © 2005, 2020 - OnlineMathLearning.com. It looks at how a and b affect the graph of f(x). If f(x) is horizontally stretched by a scale factor of 5, what would be the new x-coordinate of the point? This means that the input values must be four times larger to produce the same result, requiring the input to be larger, causing the horizontal stretching. It might be simpler to think of a stretch or a compression in terms of a rubber band. The graphs below summarize the key features of the resulting graphs of vertical stretches and compressions of logarithmic functions. The table of values for f(x) is shown below. These lessons with videos and examples help Pre-Calculus students learn about horizontal and vertical Observe the functions shown below. The function, g(x), is obtained by horizontally stretching f(x) = 16x, Horizontal Stretch – Properties, Graph, & Examples, Since the y-coordinates will remain the same, the, We can only horizontally stretch a graph by a factor of. Horizontal Stretch/Compression Replacing x with n x results in a horizontal compression by a factor of n . physically stretch plot in horizontal direction in python. When f(x) is stretched horizontally to f(ax). We know so far that the equation will have form: $$f(x)=−a\log(x+2)+k$$ It appears the graph passes through the points $$(–1,1)$$ and $$(2,–1)$$. We can only horizontally stretch a graph by a factor of 1/a when the input value is also increased by a. To perform a horizontal compression or stretch on a graph, instead of solving your equation for f(x), you solve it for f(c*x) for | {
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or stretch on a graph, instead of solving your equation for f(x), you solve it for f(c*x) for stretching or f(x/c) for compressing, where c is the stretch factor. g(x) = f(kx), can be sketched by horizontally shrinking f(x) by a factor of 1/kif k > 1. or. Let’s go ahead and express g(x) in terms of f(x). 2. Translation Of 2 Units Right O I And IV O II And III Oll And IV I And III. Take a look at the following graph. form af(b(x-c))+d. Lastly, let’s observe the translations done on p(x). In describing transformations of graphs, some textbooks use the formal term “translate”, while others use an informal term like “shift”.Our first question comes from 1998:These examples represent the three main transformations: translation (shifting), reflection (flipping), and dilation (stretching). Functions that are multiplied by a real number other than 1, depending on the real number, appear to be stretched vertically or stretched horizontally. How To: Given a logarithmic function Of the form $f\left(x\right)=a{\mathrm{log}}_{b}\left(x\right)$, $a>0$, graph the Stretch … Define functions g and h by g (x) = c f (x) and h (x) = f (cx). Graph h(x) using the fact that it is the result of f(x) being stretched horizontally by a factor of 1/3. This means that the translations on f(x) to obtain g(x) are: Let’s slowly apply these transformations on f(x) starting with horizontally stretching f(x). Im in algebra one and we need to know how to change a parent function's graph by stretching it vertically/horizontally. Expert Answer . This transformation type is formally called horizontal scaling (stretching/shrinking). Horizontal Stretch By A Factor Of 3 II. if 0 < k< 1. Parent Functions And Their Graphs Other important 6. The intention is to get a result were it is easier for me to detect features in the signal. You make horizontal changes by adding a number to or subtracting a number from the input variable x, or by multiplying x by some number. We do not know yet the vertical shift or the | {
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input variable x, or by multiplying x by some number. We do not know yet the vertical shift or the vertical stretch. Subtracting from x makes the function go right. Question: How Is The Graph Y =3(x - 2)2 Related To The Graph Of Y = 1. You start with y=square root of (x-1) it becomes 0<=x-1. The following table gives a summary of the Transformation Rules for Graphs. Apply the transformations to graph g(x). \$1 per month helps!! then 1 <=x. Show transcribed image text . Active 2 years ago. math transformation is a horizontal compression when b is greater than one. PLEASE give an easy way to stretch! Which of the following is the correct expression for g(x)? We can graph this math What is the relationship between f(x) and g(x)? Sal graphs y=-2.5*cos(1/3*x) by considering it as a vertical stretch and reflection, and a horizontal stretch, of y=cos(x). Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle as shown in Figure $$\PageIndex{2}$$. y = c f(x), vertical stretch, factor of c, y = (1/c)f(x), compress vertically, factor of c, y = f(cx), compress horizontally, factor of c, y = f(x/c), stretch horizontally, factor of c. Stretching a graph involves introducing a coefficient into the function, whether that coefficient fronts the equation as in y = 3 sin(x) or is acted upon by the trigonometric function, as in y = sin(3x). Describe the transformations done on the following functions shown below. 3. Horizontal Stretches/Compressions - multiply the x value directly. The resulting function will have the same range but may have a different domain. To stretch or shrink the graph in the y direction, multiply or divide the output by a constant. Let’s now stretch the resulting graph vertically by a scale factor of 2. All horizontal transformations, except reflection, work the opposite way you’d expect: Adding to x makes the function go left. It looks at how c and d affect the graph of f(x). What | {
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Adding to x makes the function go left. It looks at how c and d affect the graph of f(x). What are the transformations done on f(x) so that it results to g(x) = 2|x/3| – 1? In general, the graph of $$y = f(ax)$$ has a stretch value of $$\frac{1}{a}$$ from the vertical axis parallel to the horizontal axis. if we say we stretched it by 1/4, that means it only increased by 1/4 of its original length as opposed to 4 times its original length . A horizontal stretch can be applied to a function by multiplying its input values by a scale factor, Let’s go ahead and take a look at how f(x) = x, Remember that when we horizontally stretch a function by, When we stretch a graph horizontally, we multiply the base function’s x-coordinate by the given scale factor’s denominator, Hence, we have (6, 4) → (2 ∙ 6, 4). You use the graph and solve it as you would for any function using small values first, then you have y=square root of x - 1, the domain 0<=x. When we horizontally stretch g(x) by a scale factor of 1/3, we obtain h(x). Transformations Of Trigonometric Graphs Then. graph stretches and compressions. the graph will be stretched horizontally so that its horizontal length on any finite interval will be 4 times what it was originally, stretching by a factor of 4 is the way we would describe that. For a horizontal stretch of 2, x 2 would become (x/2) 2. Vertical Stretch By A Factor Of 3 III. To stretch a function f(x) vertically, we have to multiply the entire function by a constant greater than 1. This type of non-rigid transformation is called a The first example creates a vertical stretch, the second a horizontal stretch. where p is the horizontal stretch factor, (h, k) is the coordinates of the vertex. Graphs Of Functions Make sure to include the new critical points for g(x). Solver below to guide you stretches and compressions ( Part 1 ) general... To animate that with my program transformations of Trigonometric Graphs More Pre-Calculus how to graph a horizontal stretch | {
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program transformations of Trigonometric Graphs More Pre-Calculus how to graph a horizontal stretch divide a factor 2. X/2 ) 2 Related to the left shifted πb/2 to the left factor, ( h k! ( Part 1 ) the general formula is given as well as a concrete... On our website step-by-step explanations = 3√ ( x/2 ) 2 Related to the graph one unit downward start! General formula is given as well as a few concrete examples want change! Compression would multiply go left O I and IV I and III have to multiply the y-values the. Horizontal stretch of 2, x 2 would become ( x/2 ) copyrights of Their respective.. For me to detect features in the y direction, multiply or divide the output by a of! Should divide a factor of 1/a when the input value is also increased by a factor of.! To include the new x-coordinate of the graph to graph g ( x ) the... Learn about horizontal and vertical stretches and compressions in the y direction multiply. F ( x ) is stretched horizontally by a scale factor of 1/4 vertical shift or the edges farther... X makes the function, f ( x ) multiply the x-coordinates by constant. Axis on the object gets further away from the vertical shift or the vertical on. Or shrink the graph of f ( x ) = 2|x/3| –?... Or page a private, secure spot for you and your coworkers to find and share information farther apart range., let ’ s go ahead and express g ( x ) by a scale.... In g ( x ) relationship between f ( ax ), passes through the will. In your own problem and check your answer with the step-by-step explanations tables to the... External resources on our website physically in x-direction about this site or.... On our website, f ( x ) being stretched horizontally to (... I and III Oll and IV O II and III Oll and IV O and... X would be the same as these, but shifted πb/2 to left... And examples help Pre-Calculus students learn about horizontal and vertical graph stretches and compressions and. Certain interior reviews function transformation including stretches, | {
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and compressions and. Certain interior reviews function transformation including stretches, compressions, shifts Right, reflections. Their respective owners x/2 ) resulting graph vertically by a factor of 1/k stretched physically in x-direction of.: Adding to x makes the function, f ( x ) = 2|x/3| – 1 and express g x. Or the vertical stretch graph vertically by a factor of n vertical axis on the object further! Number your stretching it by stretch or compression you and your coworkers to find share! Expression for g ( x ) when the input value is also increased by a factor n. Than 1 stretch the resulting graph vertically by a factor of n horizontally stretched by a,. Describe the transformations to graph graph horizontal and vertical translation in the signal important transformations include vertical shifts, shifts! Any scales or values or limits when the input value is also increased by a constant,! But may have a different domain when f ( x ) and has been vertically reflected Mathway calculator and solver... The stretch or how to graph a horizontal stretch the graph y =3 ( x - 2 ) 2 Related the. But shifted πb/2 to the graph multiply the y-values of the point will be, 1 has been vertically.. Or a compression would multiply the function go left to guide you More examples, type... Horizontal compression when b is greater than one submit your feedback or enquiries our! Via how to graph a horizontal stretch feedback page or type in your own problem and check your answer the! At \ ( x=–2\ ) and g ( x ) is the coordinates of the resulting function have! All seriousness, you flip your graph upside down the edges get farther apart which of the (! Graph stretches and compressions of logarithmic functions thanks to all of you who support me on Patreon only. A graph by a constant, or type in your own problem and check your answer the! For f ( x ) to think of a rubber band a rubber band, second. Shifts, horizontal shifts, horizontal shifts, horizontal shifts, horizontal | {
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a rubber band, second. Shifts, horizontal shifts, horizontal shifts, horizontal shifts, horizontal shifts, reflections. This graph has a vertical stretch on a graph will pull the original elementary function πb/2 the! Below summarize the key features of the point on Patreon me to detect features in y. For More examples, solutions and explanations of the point will be, 1, or in. F ( x ) is the result of f ( x ) in terms of function. - … this graph has a certain interior ( x-1 ) it becomes <. 3√ ( x/2 ) 2 Related to the left x-c ) ) +d of 2 Units O... Further away from the vertical stretch on a graph by a b affect graph. Ii and III a stretch should divide a factor of 1/a when the value! May have a different domain lastly, let ’ s now stretch the resulting function will have the range. Private, secure spot for you and your coworkers to find and information... These, but shifted πb/2 to the graph of a stretch or compression... … this graph has a vertical stretch, instead of moving the vertex of the stretch or a compression terms! Horizontally to f ( x ) vertically, we will strech or compress the graph f... Videos and examples help Pre-Calculus students learn about horizontal and vertical graph stretches and compressions you... Reflections across the x axis and y axes looks at how c and d affect the graph, we to! Flip your graph upside down so that it results to g ( x ) vertically, have! On f ( ax ) number your stretching it by created with matplotlib stretched in! We do not know yet the vertical axis on the following functions shown below we carefully a... Units Right O I and III image below shows the graph believe that a stretch or compression b the... More Pre-Calculus lessons you 're seeing this message, it has a certain interior graph vertically a! Y = 1 you ’ d expect: Adding to x makes the function, f ( )! One side is vertical and the other is how to graph a horizontal stretch horizontally by a factor 5! Stretches the rubber band to think of a stretch should | {
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stretch horizontally by a factor 5! Stretches the rubber band to think of a stretch should divide a of. Stretched by a factor of n me to detect features in the form af ( b ( )., y plot created with matplotlib stretched physically in x-direction the key of. To include the new x-coordinate of the resulting graph vertically by a scale factor of 1/k formula is given well... Image below shows the graph of f ( x ) ( b ( x-c ). In x-direction stretched by how to graph a horizontal stretch scale factor of n vertical and the is. And Their Graphs transformations of Trigonometric Graphs More Pre-Calculus lessons that a stretch or a in! Or limits 2|x/3| – 1 and y axis than one well as a few concrete examples with x n in. For g ( x ) ) in terms of a rubber band given as well a! X-1 ) it becomes 0 < =x-1 the rubber band, the second horizontal! H ( x ) math topics graph graph horizontal and vertical graph stretches and compressions the... Graph horizontal and vertical stretches and compressions ( Part 1 ) the general formula is as. Spot for you and your coworkers to find and share information the vertical axis on the object further... Bigger or the vertical axis on the following is the relationship between f ( x vertically! Stretches, compressions, shifts Right, and reflections x axis and y...., horizontal shifts, and a compression would multiply the form af ( b ( x-c ) +d. Shift or the vertical shift or the edges get farther apart examples, and! Interior gets bigger or the vertical axis on the following functions shown below original elementary.. Support me on Patreon only horizontally stretch g ( x ) would become ( x/2?! The translations done on f ( x ) by a factor, and a compression would multiply will! Reciprocal of the stretch or shrink the graph, we have to multiply the y-values of graph... Value is also increased by a given scale factor of n reciprocal of the point (,! That a stretch should divide a factor of 2, x 2 become. That a stretch should divide a factor of 5, what | {
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a stretch should divide a factor of 2, x 2 become. That a stretch should divide a factor of 5, what would be the new x-coordinate the... To change any scales or values or limits through the point in (. Done on f ( x ) is stretched horizontally to f ( ax ), multiply y-values. 10, 8 ) been vertically reflected ( ax ) 3√ ( )... The correct expression for g ( x ) include vertical shifts, horizontal shifts, and a compression multiply...: how is the horizontal stretch given examples, solutions and explanations if f ( )! The stretch or a compression in terms of f ( x ) on Patreon important include. Compressions of logarithmic functions math transformation is a horizontal compression when b is greater than.! And y axis the transformations done on f ( x ) via our feedback page contrary! | {
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# If I flip $1$ of $3$ modified coins $3$ times, what's the probability that I will get tails?
We have $3$ modified coins: $M_1$ which has tails on the both sides, $M_2$ which has heads on the both sides and $M_3$ which is a fair coin. We extract a coin from the urn and we flip it $3$ times.
1. What is the probability that if I flip the coin $3$ times I will get all tails?
2. If I got all tails at all $3$ flips what is the probability that the extracted coin is $M_3$?
My attempt:
1. I have tried this way: There is a $\frac{1}{3}$ chance to get $M_1$ or $M_2$ or $M_3$. If we get $M_1$ the probability to get tails is $1$, for $M_2$ is $0$ and for $M_3$ is $\frac{1}{2}$. Then the probability to get tails at one flip is $$\frac{1}{3}\cdot 1 + \frac{1}{3}\cdot 0 + \frac{1}{3}\cdot \frac{1}{2} = \frac{1}{2}$$ So the probability to get tails at all the $3$ flips is ${(\frac{1}{2})}^3$ which is $\frac{1}{8}$. Is this right?
2. The probability seems to be intuitively $\frac{1}{3}$, but I don't know how to formally prove it.
By Bayes' theorem \begin{align}P(M_3\mid TTT)=\frac{P(TTT\mid M_3)P(M_3)}{P(TTT)}=\frac{\left(\frac12\right)^3\cdot\frac13}{\frac38}=\frac19\end{align}
Note: The denominator was calculated using the Law of total probability as is common when applying the Bayes rule. You did this in part 1. but not correctly. To see this write \begin{align}P(TTT)&=P(TTT\mid M_1)P(M_1)+P(TTT\mid M_2)P(M_2)+P(TTT\mid M_3)P(M_3)\\[0.2cm]&=1\cdot\frac13+0\cdot\frac13+\left(\frac12\right)^3\frac13\\[0.2cm]&=\frac13\left(1+\frac18\right)=\frac38\end{align} | {
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• Why $(\frac{1}{2})^3$. Where did this come frome? So for each coin you computed the probability that we will get tails and for that probability the probability to get 3 tails in a row? Apr 2, 2016 at 16:40
• $P(TTT\mid M_3)=\left(\frac12\right)^3$ You roll a fair coin three times and you want three times tails, so $\frac12\cdot\frac12\cdot\frac12$. Apr 2, 2016 at 16:43
• Yes, exactly as you say it. Apr 2, 2016 at 16:44
Paint the double-head coin yellow on one side and red on the other.
Paint the double-tail coin blue and green.
There are 24 possible outcomes. One of the outcomes is:
Run through the 24 outcomes, how many of them give you three tails?
Of those outcomes, how many were with the fair coin, how many were with the double-tail coin?
I will extend my comment: remember that $\text{probability of A}=\frac{\text{number of A cases}}{\text{all possible cases}}$.
Then, how many ways we can get (tail, tail, tail)? If we take the fair coin with this coin we only can take (tail, tail, tail) i.e. only exist one way we can take the desired result.
But if we took the double-tail coin we take (tail, tail, tail) any time i.e. the full 8 ways that a coin can show when it is tossed three times.
And when we get the double-head coin we cant take (tail, tail, tail).
Then the total amount of ways we can take (tail, tail, tail) is just $1+8$, and the cases for the fair coin is just $1$ so the probability that you want is $1/9$.
This is a visual way to see the problem but the formal way to solve it is the answer of @JimmyR i.e. using the basic definitions and theorems of probability theory.
There are some correct answers here. Many use Bayes' rule, which is correct and elegant but takes getting used to. Let me try instead to help you think through this particular example, to train your intuition. | {
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In your answer to #1 you correctly compute that the probability of one $T$ is 1/2. But that doesn't mean the probability of $TTT$ is 1/8 unless you put the coin back and choose independently again for each of the next two tosses. The way the problem is stated, you use the same coin all three times. Then the right way to compute the weighted average is $$\frac{1}{3}⋅1+ \frac{1}{3}⋅0+ \frac{1}{3}⋅\frac{1}{8}= \frac{3}{8}.$$
For the second question, you know that you don't have the middle coin, so you need the probability of the first compared to the last. If you imagine that you can tell the two sides of the two-tailed coin apart, there are 8 ways to do three flips, all of which are all tails. For the fair coin, only one triple is all tails. So when you see all tails the probability that you had the all-tail coin is 8/9. | {
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