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linear function : a function with a constant rate of change that is a polynomial of degree 1, and whose graph is a straight line | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-terms |
model breakdown : when a model no longer applies after a certain point | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-terms |
parallel lines : two or more lines with the same slope | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-terms |
perpendicular lines : two lines that intersect at right angles and have slopes that are negative reciprocals of each other | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-terms |
point-slope form : the equation for a line that represents a linear function of the formyây1=m(xâx1)yây1=m(xâx1) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-terms |
slope : the ratio of the change in output values to the change in input values; a measure of the steepness of a line | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-terms |
slope-intercept form : the equation for a line that represents a linear function in the formf(x)=mx+bf(x)=mx+b | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-terms |
vertical line : a line defined byx=a,x=a,whereaais a real number. The slope of a vertical line is undefined. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/4-key-terms |
A polynomial function of degree two is called a quadratic function. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The axis of symmetry is the vertical line passing through the vertex. The zeros, orx-x-intercepts, are the points at which the parabola crosses thex-x-axis. They-y-intercept is the point at which the parabola crosses they-y-axis. SeeExample 1,Example 7, andExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The vertex can be found from an equation representing a quadratic function. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The domain of a quadratic function is all real numbers. The range varies with the function. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
A quadratic functionâs minimum or maximum value is given by they-y-value of the vertex. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue. SeeExample 5andExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The vertex and the intercepts can be identified and interpreted to solve real-world problems. SeeExample 9. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
A power function is a variable base raised to a number power. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The end behavior depends on whether the power is even or odd. SeeExample 2andExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function. SeeExample 6andExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
A polynomial of degreennwill have at mostnnx-intercepts and at mostnâ1nâ1turning points. SeeExample 8,Example 9,Example 10,Example 11, andExample 12. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Polynomial functions of degree 2 or more are smooth, continuous functions. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero. SeeExample 2,Example 3,andExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Another way to find thex-x-intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses thex-x-axis. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The multiplicity of a zero determines how the graph behaves at thex-x-intercepts. SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The end behavior of a polynomial function depends on the leading term. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The graph of a polynomial function changes direction at its turning points. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
A polynomial function of degreennhas at mostnâ1nâ1turning points. SeeExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at mostnâ1nâ1turning points. SeeExample 8andExample 10. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Graphing a polynomial function helps to estimate local and global extremas. SeeExample 11. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The Intermediate Value Theorem tells us that iff(a)andf(b)f(a)andf(b)have opposite signs, then there exists at least one valueccbetweenaaandbbfor whichf(c)=0.f(c)=0.SeeExample 9. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree. SeeExample 1andExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the formxâk.xâk.SeeExample 3,Example 4,andExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Polynomial division can be used to solve application problems, including area and volume. SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
To findf(k),f(k),determine the remainder of the polynomialf(x)f(x)when it is divided byxâk.xâk.This is known as the Remainder Theorem. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
According to the Factor Theorem,kkis a zero off(x)f(x)if and only if(xâk)(xâk)is a factor off(x).f(x).SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
According to the Rational Zero Theorem, each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient. SeeExample 3andExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
When the leading coefficient is 1, the possible rational zeros are the factors of the constant term. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Synthetic division can be used to find the zeros of a polynomial function. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
According to the Fundamental Theorem, every polynomial function has at least one complex zero. SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Every polynomial function with degree greater than 0 has at least one complex zero. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form(xâc),(xâc),whereccis a complex number. SeeExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The number of negative real zeros of a polynomial function is either the number of sign changes off(âx)f(âx)or less than the number of sign changes by an even integer. SeeExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division. SeeExample 9. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
We can use arrow notation to describe local behavior and end behavior of the toolkit functionsf(x)=1xf(x)=1xandf(x)=1x2.f(x)=1x2.SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Application problems involving rates and concentrations often involve rational functions. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The domain of a rational function includes all real numbers except those that cause the denominator to equal zero. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero. SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
A rational functionâs end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. SeeExample 7,Example 8,Example 9, andExample 10. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior. SeeExample 11. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
If a rational function hasx-intercepts atx=x1,x2,â¦,xn,x=x1,x2,â¦,xn,vertical asymptotes atx=v1,v2,â¦,vm,x=v1,v2,â¦,vm,and noxi=anyvj,xi=anyvj,then the function can be written in the form | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
SeeExample 12. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
The inverse of a quadratic function is a square root function. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Iffâ1fâ1is the inverse of a functionf,f,thenffis the inverse of the functionfâ1.fâ1.SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
While it is not possible to find an inverse of most polynomial functions, some basic polynomials are invertible. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
To find the inverse of certain functions, we must restrict the function to a domain on which it will be one-to-one. SeeExample 3andExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
When finding the inverse of a radical function, we need a restriction on the domain of the answer. SeeExample 5andExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Inverse and radical and functions can be used to solve application problems. SeeExample 6andExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
A relationship where one quantity is a constant multiplied by another quantity is called direct variation. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Two variables that are directly proportional to one another will have a constant ratio. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
A relationship where one quantity is a constant divided by another quantity is called inverse variation. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
Two variables that are inversely proportional to one another will have a constant multiple. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-concepts |
f ( x ) = a x 2 + b x + c f ( x ) = a x 2 + b x + c | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-equations |
f ( x ) = a ( x â h ) 2 + k f ( x ) = a ( x â h ) 2 + k | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-equations |
f ( x ) = a n x n + ... + a 2 x 2 + a 1 x + a 0 f ( x ) = a n x n + ... + a 2 x 2 + a 1 x + a 0 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-equations |
f ( x ) = d ( x ) q ( x ) + r ( x ) where q ( x ) â 0 f ( x ) = d ( x ) q ( x ) + r ( x ) where q ( x ) â 0 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-equations |
f ( x ) = P ( x ) Q ( x ) = a p x p + a p â 1 x p â 1 + ... + a 1 x + a 0 b q x q + b q â 1 x q â 1 + ... + b 1 x + b 0 , Q ( x ) â 0 f ( x ) = P ( x ) Q ( x ) = a p x p + a p â 1 x p â 1 + ... + a 1 x + a 0 b q x q + b q â 1 x q â 1 + ... + b 1 x + b 0 , Q ( x ) â 0 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-equations |
y = k x n , k y = k x n , k is a nonzero constant. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-equations |
y = k x n , k y = k x n , k is a nonzero constant. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-equations |
arrow notation : a way to represent symbolically the local and end behavior of a function by using arrows to indicate that an input or output approaches a value | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
axis of symmetry : a vertical line drawn through the vertex of a parabola, that opens up or down, around which the parabola is symmetric; it is defined byx=âb2a.x=âb2a. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
coefficient : a nonzero real number multiplied by a variable raised to an exponent | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
constant of variation : the non-zero valuekkthat helps define the relationship between variables in direct or inverse variation | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
continuous function : a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
degree : the highest power of the variable that occurs in a polynomial | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
Descartesâ Rule of Signs : a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes off(x)f(x)andf(âx)f(âx) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
direct variation : the relationship between two variables that are a constant multiple of each other; as one quantity increases, so does the other | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
Division Algorithm : given a polynomial dividendf(x)f(x)and a non-zero polynomial divisord(x)d(x)where the degree ofd(x)d(x)is less than or equal to the degree off(x)f(x), there exist unique polynomialsq(x)q(x)andr(x)r(x)such thatf(x)=d(x)q(x)+r(x)f(x)=d(x)q(x)+r(x)whereq(x)q(x)is the quotient andr(x)r(x)is the remaind... | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
end behavior : the behavior of the graph of a function as the input decreases without bound and increases without bound | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
Factor Theorem : kkis a zero of polynomial functionf(x)f(x)if and only if(xâk)(xâk)is a factor off(x)f(x) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
Fundamental Theorem of Algebra : a polynomial function with degree greater than 0 has at least one complex zero | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
general form of a quadratic function : the function that describes a parabola, written in the formf(x)=ax2+bx+cf(x)=ax2+bx+c, wherea,b,a,b,andccare real numbers andaâ0.aâ0. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
global maximum : highest turning point on a graph;f(a)f(a)wheref(a)â¥f(x)f(a)â¥f(x)for allx.x. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
global minimum : lowest turning point on a graph;f(a)f(a)wheref(a)â¤f(x)f(a)â¤f(x)for allx.x. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
horizontal asymptote : a horizontal liney=by=bwhere the graph approaches the line as the inputs increase or decrease without bound. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
Intermediate Value Theorem : for two numbersaaandbbin the domain off,f,ifa<ba<bandf(a)âf(b),f(a)âf(b),then the functionfftakes on every value betweenf(a)f(a)andf(b)f(b); specifically, when a polynomial function changes from a negative value to a positive value, the function must cross thex-x-axis | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
inverse variation : the relationship between two variables in which the product of the variables is a constant | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
inversely proportional : a relationship where one quantity is a constant divided by the other quantity; as one quantity increases, the other decreases | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
invertible function : any function that has an inverse function | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
joint variation : a relationship where a variable varies directly or inversely with multiple variables | https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-key-terms |
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