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income tax you pay depends on your income. b. Suppose a rock is dropped straight down from a high place. Physics feet. c. The weather bureau records the temperature over a 24-hour period tells us that the distance traveled by the rock in t seconds is 16t2 in the form of a graph (Figure 3.1-1). The graph shows the temp... |
uating a Function Find the indicated values of a. f 3 2 1 Solution b. f 1 2x2 1. x f 2 1 5 2 c. f 0 1 2 a. To find the output of the function f for input 3, simply replace x with 3 in the function rule and simplify the result. 232 1 210 3.162 5 2 1 226 5.099 Similarly, replace x with 5 f and 0 for b and c. 2 5 3 f 2 1 ... |
, then exactly one value of y is produced. So the equation defines a function whose domain is the set of all real numbers and whose rule is stated below. f x 2 y2 x 1 0 1 3 2x 5 A 2 can not be solved uniquely for y: b. The equation y2 x 1 y Β± 2x 1 or y 2x 1 y 2x 1 This equation does not define y as a function of x beca... |
1 2 A real-life situation may lead to a function whose domain does not include all the values for which the rule of the function is defined. 146 Chapter 3 Functions and Graphs Example 7 Finding the Domain of a Profit Function A glassware factory has fixed expenses (mortgage, taxes, machinery, etc.) of $12,000 per week... |
function is a piecewise-defined function with infinitely many pieces if 3 x 6 2 if 2 x 6 1 if 1 x 6 0 if 0 x 6 1 if 1 x 6 2 if 2 x 6 3 The rule can be written in words as follows: For any number x, round down to the nearest integer less than or equal to x. The domain of the greatest integer function is all real number... |
. 40. 42 10x x 2 x 3 1 x In Exercises 5β12, determine whether the equation defines y as a function of x. In Exercises 43β56, determine the domain of the function according to the domain convention. 5. y 3x 2 12 6. y 2x 4 3x 2 2 7. y 2 4x 1 8. 5x 4y 4 64 0 9. 3x 2y 12 10. y 4x 3 14 0 11. x2 y 2 9 12. y2 3x 4 8 0 43. f x... |
function of its a. side s b. diagonal d 65. A box with a square base of side x is four times higher than it is wide. Express the volume V of the box as a function of x. 66. The surface area of a cylindrical can of radius r 2pr 2 2prh. and height h is If the can is twice as high as the diameter of its top, express its ... |
function that represents the income tax law. What is the domain of the function? 150 Chapter 3 Functions and Graphs 72. The table below shows the 2002 federal income a. Write a piecewise-defined function T such that x T x is the tax due on a taxable income of dollars. What is the domain of the function? 2 1 b. Find T ... |
f 0 b. To find f(3), notice that the point (3, 0) is on the graph. Thus, 0 is the output produced by the input 3, or f 3 7. 0. 1 1 2 2 c. To find the domain of the function, find the x-coordinates of the point farthest to the left and farthest to the right. Then, determine whether there are any gaps in the function be... |
2., There is no vertical line that intersects the graph above in more than one place, so this graph defines a function. β Analyzing Graphs In order to discuss a graph or compare two graphs, it is important to be able to describe the features of different graphs. The most important features are the x- and y-intercepts,... |
.7 Example 4 Finding Local Maxima and Minima 3.1 Figure 3.2-7 Graph f x 2 1 Solution x 3 1.8x2 x 1 and find all local maxima and minima. 1.2 0.1 1.1 Figure 3.2-8 In the decimal or standard window, the graph does not appear to have any local maxima or minima (see Figure 3.2-7). Select a viewing window such as the one in... |
and, 1 2 2 c.βd. The function is concave up on the left and concave down on the right, and the inflection point appears to be at about Thus, the funcand concave down over the tion is concave up over the interval interval q, 1 x 1. 1, q. 1 2 1 2 β 10 4.7 4.7 10 Figure 3.2-11 Section 3.2 Graphs of Functions 155 Graphs o... |
Y1 Y2 6, 1 X2, 3 X 2, 4 1, 4 3 4 How does your graph compare with Figure 3.2-12? Example 7 The Absolute-Value Function 3.1 Graph f x 1 2 x. 0 0 Solution 4.7 4.7 3.1 Figure 3.2-13 The absolute-value function tion, since by definition f x 2 1 x 0 0 is also a piecewise-defined func- x 0 0 x x e if x 6 0 if x 0 Its graph ... |
graph are each given as a function of a third variable, t, called a parameter. The functions that give the rules for the coordinates are called parametric equations. Technology Tip To change to dot mode, select DOT in the TI MODE menu. In the Casio SETUP menu, set the DRAWTYPE to PLOT. 158 Chapter 3 Functions and Grap... |
let x t y f(t) To graph x f(y) in parametric mode, let x f(t) y t Example 10 Graphing in Parametric Mode Graph the following equations in parametric mode on a calculator. a. y 2 x 1 2 a b 3 Solution b. x y2 3y 1 a. Let x t and y 2 t 1 2 b a 3. b. Let x t2 3t 1 and y t. 10 10 10 10 10 10 10 10 Figure 3.2-17 Notice that... |
. Find the approximate intervals on which the function is increasing, decreasing, and constant. 17. f 18. g 19 8x 2 8x 5 Section 3.2 Graphs of Functions 161 20. f 21. g x x 1 1 2 2 22. g x 2 1 x 4 0.7x 3 0.6x 2 1 0.2x 4x 2 x 1 In Exercises 23β28, graph each function. Estimate all local maxima and minima of the function... |
on which the function is increasing, decreasing, and constant. c. Estimate all local maxima and minima. d. Find the approximate intervals on which the function is concave up and concave down. e. Estimate the coordinates of any inflection points. 37. f 39 2x 1 38. f x 3 3x 2 2 40. g x 2 4x 3 x 3 4x 2 x x 1 1 2 2 In Exe... |
x 2 4 3x 2 In Exercises 60 and 61, draw the graph of a function f that satisfies the given conditions. The function does not need to be given by an algebraic rule. 1 2 2 2 60 when x is in the interval 1, a 1 2b starts decreasing when 3 f starts increasing when 0 1 2 x 1 x 5 61. β’ domain β’ range 1 β’ 1 2b, 4 3 5, 6 3 3 4... |
minimum β’ always has exactly 1 y-intercept β’ can have 0, 1, or 2 x-intercepts A parabola is symmetric about a line through the vertex called the axis of symmetry. Quadratic Functions Quadratic functions can be written in several forms. 164 Chapter 3 Functions and Graphs Three Forms of a Quadratic Function A quadratic ... |
intercept is c ax2 bx c The x-intercepts are the solutions of the quadratic equation 0, which can be solved either by factoring or by the quadratic formula. In general, the x-intercepts are b 2b2 4ac 2a and b 2b2 4ac 2a for values which make one x-intercept. If b2 4ac b2 4ac positive. If b2 4ac 0, is negative, then the... |
the graph of a quadratic function has no x-intercepts, then there are no real values of s and t for which 21 2 x-Intercept Form The x-intercept form is the most useful form for finding the x-intercepts. For function written form, in x f the x-intercepts of the graph are s and t. Notice f that both of these values are ... |
and x-Intercept Form Write the following functions in polynomial and x-intercept form, if possible. a. b. c.4 3x 2 3.9x 43.2 2 x 2 x 4 2 21 2 1 Solution a. To change the function f to polynomial form, distribute and collect like terms 6x 9 0.4 0.4 0.4x 2 2.4x 5.6 1 2 2 f b2 4ac 3.2 Since the function f cannot be writt... |
by the leading coefficient or add a term to both sides. It is necessary to factor out the leading coefficient, then add and subtract the 2 term. b 2b a Summary of Quadratic Functions Section 3.3 Quadratic Functions 169 b. Multiply the terms in parentheses, then complete the square as shown below. 0.3 2 0.3 x 1 1 g x 1... |
opens downward, so the maximum occurs at the vertex. The function is in polynomial form, so the vertex occurs when x b 2a 1500 1 2 2 1 562,500 ft2, 750. The largest possible area is y 1500 750 750. x 750 The dimensions of the field are 750 by 750 ft. which occurs when and β Example 7 Maximizing Profit A vendor can sel... |
9. f 10. h x x 1 1 2 2 11. g 12 21 x 3 1 x 3 2 x 1 21 2 x 1 2b 4b a 0.4 x 2.1 x 0.7 2 21 1 In Exercises 13β21, determine the vertex and x- and y-intercepts of the given quadratic function, and sketch a graph. 13 14. g 15. 17. 19. 21 8x 2 1 2 x 1 21 x 3 x 3 2 x 4 1 21 16. 18. 20 6x 3 2x 2 4x 21 x 6 2 Write the followin... |
side by a river is to be fenced on three sides to form a rectangular enclosure. If the total length of fence is 200 feet, what dimensions will give an enclosure of maximum area? 46. A salesperson finds that her sales average 40 cases per store when she visits 20 stores per week. If she visits an additional store per w... |
is it at that time? (i) y 52. A ball is thrown upward from the top of a 96-ft tower with an initial velocity of 80 ft/sec. When does the ball reach its maximum height and how high is it at that time? x x (ii) y x x (iii) (iv) 3.4 Graphs and Transformations Objectives β’ Define parent functions β’ Transform graphs of par... |
x) = x Square root function Figure 3.4-1 4 2 β2 β2 β4 (1, 1) 2 (0, 0) 4 x 3 f(x) = x Cube root function The parent functions will be used to illustrate the rules for the basic transformations. Remember, however, that these transformation rules work for all functions. 174 Chapter 3 Functions and Graphs Technology Tip If... |
x x 1 x2 (x 1)2 3 4 9 16 16 9 5 4 25 16 When 1 is subtracted from the x-values, the result is that the entries in the table shift 1 position to the right. Thus, the entire graph is shifted 1 h unit to the right. Construct a table for to see that the entries shift 3 positions to the left. x 3 x 1 2 2 1 2 Horizontal Shi... |
across the y-axis, as shown in Figure 3.4-4. 1x a, b x x f 2 1 2 1 Technology Tip If the function f is entered as Y1, then the function g x f x 1 2 can be entered in Y2 as Y1 and the function 1 2 g x f x 1 2 can be entered in Y3 as Y1(X). 1 2 y (βa, b) (a, b) x (a, βb) Figure 3.4-4 Section 3.4 Graphs and Transformatio... |
ching and Compressing a Graph Graph g x 2 1 2x 3 and h 1 4 x 1 2 x 3. Solution every The parent function is y-coordinate of the parent function is multiplied by 2, stretching the graph For the function x g x f 2 1 2 1 2x 3, x 3. of the function in the vertical direction, away from the x-axis. For the func- 1 4 x 2 1 x ... |
is a point on the graph of f(x), then point on the graph of If c 77 1, the graph of g(x) f(c x). g(x) f(c x) is the graph of f 1 c x, y b a is a compressed horizontally, toward the y-axis, by a factor of 1 c. If c 66 1, the graph of g(x) f(c x) horizontally, away from the y-axis, by a factor of is the graph of f stret... |
) x y 4 2 (1, 1) x 0 β4 β2 β2 2 (0, 0) 4 β4 0 β4 β2 β2 β4 4 2, 0) x 2 4 (1, β1) 0 β4 β2 β2 β4 y 1 2 (, 4) (1, 3) 4 2 0 β4 β2 β2 β4 2 4 Figure 3.4-8 x β Section 3.4 Graphs and Transformations 181 Graphing Exploration 1 2 1 2 x g x 2x 322x 4 1, to produce the graph of For the function list several different possible orde... |
parent function: f 1 x x 2 1 6. f x 1 2 3x 2 transformations: shift the graph 2 units to the right, stretch it horizontally by a factor of 2, and shift it upward 2 units 7. 8 223 x 5 9 In Exercises 10β21, graph each function and its parent function on the same set of axes. 10. f x 1 2 x 2 11. h x 2 1 1 x 12. 14. 16 2x... |
g x x 1 1 2 2 3f 1 β1 42. 44.25f x 1 2 In Exercises 46β49, use the graph of the function f in the figure to sketch the graph of the function h. Use the graph of f to sketch the graph of the function g. y f(x) x 56. 58. 60. g g g 61 57. 59. g g x x 1 1 2 2 21 x 2 4 321 x 2 21 x 2 21 21 46. 48 47. 49 2x 2 In Exercises 5... |
+ 4y2 = 24y (β6, 3) P (βx, y) (6, 3) Q (x, y) x Figure 3.4.A-1 Each point P on the left side of the graph has a mirror image point Q on the right side of the graph, as indicated by the dashed lines. Note that β’ their y-coordinates are the same β’ their x-coordinates are opposites of each other β’ the y-axis is the perpe... |
graph, then is also on it. In algebraic terms, this means that replacing y by produces an equivalent equation. y 186 Chapter 3 Functions and Graphs Example 2 x-Axis Symmetry Verify that y2 4x 12 is symmetric with respect to the x-axis. NOTE 0 x Except for, the graph of a 1 2 f function is never symmetric with respect ... |
the graph is symmetric with respect to the origin. β Graphing Exploration Graph the equation y x 3 10 x from Example 3. Use the TRACE feature to locate at least three points (x, y) on the graph and show that for each point, is also on the graph. x, y 1 2 Summary There are a number of techniques used to understand the ... |
etry 189 A function f is odd if f(x) f(x) for every value x in the domain of f. The graph of an odd function is symmetric with respect to the origin. For example, f x 2 1 x 3 2x is an odd function because x x 3 2x x x 3 2x 2x 2 1 Hence, the graph of f is symmetric with respect to the origin, as you can verify with your... |
4), (5, 0), 6, 1,. 4, 1 b. Suppose the points in part a lie on the graph of 1, 1 7, 3 and,, 2 1 2 1 1 2 1 4 an odd function f. Plot the points 1 7 3, f 5 5, f 22 1 (4, f(4)), and (6, f(6)). 7, f 3 22,, 1 1 1 1 1 2 2, f, 22 2, 22 (1, f(1)), 1 40. a. Plot the points 1,3 1, 2 1 5, 2, 2 1 3, 5 2, 3, 2 1 2, and 4, 1. 2 1 1... |
different letter h for the sum function, we shall usually f g denote it by f is defined by the rule g Thus, the sum f g 21 2 1 2 1 2 NOTE This rule is not just a formal manipulation of symbols. If x is and g(x). The plus sign in a number, then so are addition of numbers, and the result is a number. But the plus sign i... |
otient Functions For f x 1 2 23x and g 2x2 1 x 1 2 a. write the rule for fg and f g. b. find the domain of fg and f g. Solution a. fg x 2 21 1 f gb1 x 2 a 23x 2x 2 1 23x 3 3x 23x 23x 3 3x x 2 1 2x 2 1 b. The domain of f consists of all x such that 3x 0, that is, x 0. x2 1 0, Similarly, the domain of g consists of all x... |
21 21 Solution b. d. f g f g 1 x 2 21 21 2 1 1 a. To find g f 1 2 2 21, first find 17 2 1 2 Next, use the result as an input in g. So g f 2 1 b. To find 2 21 f g 1 1 19 1 21 17 g 1 2 1 17 2 1 19. 2, first find Next, use the result as an input in f. So 1 f g 1 2 21 5 194 Chapter 3 Functions and Graphs c. To find x 21 g... |
Operations on Functions 195 Expressing Functions as Composites In calculus, it is often necessary to write a function as the composition of two simpler functions. For a function with a complicated rule, this can usually be done in several ways. Example 5 Writing a Function as a Composite 33x 2 1. h x Let 1 ferent ways... |
the area function 21 A 18 2t 3 b a p 2 18 2t 3b a At t 12 minutes, the area of the surface of the puddle is A r 1 12 2 21 p 18 2 12 3b a 2 4p 9 1.396 square inches., and t 22 β Exercises 3.5 In Exercises 1β4, find and their domains. ( f g)(x), ( f g)(x), (g f )(x), In Exercises 8β11, find the domains of fg and f g. 1.... |
In Exercises 23β26, find the rules of the functions ff and f f. 23. 25. f f x x 1 1 2 2 x3 1 x 24. 26 39. In Exercises 27β30, verify that (g f )(x) x for the given functions f and g. (f g)(x) x and 27. 28. 29. 30 9x 2 g x 2 9 x 1 2 23 x 1 g 1 23 41 2x 3 5 g x 2 1 In Exercises 31β36, write the given function as the com... |
βs shadow as a function of time. Hint: first use similar triangles to express s as a function of the distance d from the streetlight to Brandon. g(x) In Exercises 48β52, let and the composite function graph.. x 0 0 f g f Graph the function f on the same x A 00 00 B 48. 50.5 x 4 2 9 2 1 49. 51. f f 52. Write a piecewise... |
Section 3.5.A Excursion: Iterations and Dynamical Systems 199 3.5.A Excursion: Iterations and Dynamical Systems Discrete dynamical systems, which deal with growth and change, occur in economics, biology, and a variety of other scientific fields. Compound interest provides a simple example of a discrete dynamical syste... |
calculator. Technology Tip If a function has been entered as Y1 in the equation memory, it can be iterated as follows: β’ Store the initial value as X. β’ Key in Y1 STO X. Pressing ENTER repeatedly produces the iterated values of the function. Example 1 Iterated Function Notation Write the first four iterations of the f... |
and that the orbit appears to converge to 0. Now let x 2. Figure 3.5.A-3 shows that the orbit begins 2, 4, 16, 256, 65536, 4294967296,... In this case, the orbit is not converging: its terms get larger and larger without bound as n increases. The fact is expressed by saying that the orbit diverges or that the orbit ap... |
. 2 B B B 1 Therefore, the orbit is Because the orbit begins to repeat its values after a few steps, it is said to be eventually periodic and is said to be an eventually periodic point. 22 β Example 5 Orbit Analysis Analyze all the orbits of f x2 x 1 2 and illustrate them graphically. Solution Example 3 and the Calcula... |
x 2 19. Determine all the fixed points and all the x eventually fixed points of the function. Find the orbit of x for every integer value of x such that fixed, eventually fixed, periodic, or eventually periodic.. Classify each point as b. Describe a pattern in the classifications you made in part a. 21. Let x 1 0 2 x ... |
that is a point on its inverse function or relation. This fact can be used to graph the inverse of a function. is a point on the graph of a function. Then y, x 2 1 2 1 Example 1 Graphing an Inverse Relation The graph of a function f is shown below. Graph the inverse, and describe the relationship between the function ... |
.2x 3 2x 0.5 and its Solution To graph f, let x1 t and y1 0.7t5 0.3t4 0.2t3 2t 0.5. To graph x2 the inverse of 0.7t5 0.3t4 0.2t3 2t 0.5 f, exchange x y2 t. and and y by letting Figure 3.6-6 shows the graph of f and its inverse on the same screen with the graph of y x. Section 3.6 Inverse Functions 207 3.1 4.7 4.7 3.1 F... |
One-to-One Functions A function f is one-to-one if f(a) f(b) implies that a b. NOTE By the definition implies of a function, b a that f f 1 2 1 2 a b. If a function is one-to-one, then its inverse is also a function. Determining Whether a Graph is One-to-One In Example 1, the points 1 These two points have different i... |
to-one. The function and its inverse are shown in Figure 3.6β9. β NOTE The function f in Example 5 is always increasing and the function h is always decreasing. Every increasing or decreasing function is one-to-one because its graph can never touch the same horizontal line twiceβit would have to change from increasing ... |
2 a 1 to the result, you 2 f Similarly, 1 f f a 1 22 22 f a 1 2 b The results above can be generalized to all values in the domains of f and 1. f Composition of Inverse Functions A one-to-one function f and its inverse function these properties. f 1 have ( f 1 f )(x) x for every x in the domain of f ( f f 1) (x) x for... |
1 In Exercises 9 β 22, find the rule for the inverse of the given function. Solve your answers for y and, if possible, write in function notation (see Examples 3 and 4). 9. 11. 13. 15. 17 5x 2 4 5 2x 3 24x 7 1 x 10. 12. 14. 16. 18 3x 2 5 x5 1 1 3 2 5 23x 2 1 2x 19. f x 1 2 1 2x 2 1 20. f x 1 2 x x 2 1 21 22. f x 1 2 5... |
function of f. 214 Chapter 3 Functions and Graphs 52. List three different functions (other than the one in Exercise 51), each of which is its own inverse. Many correct answers are possible. 53. Critical Thinking Let m and b be constants with is m 0. one-to-one, and find the rule of the inverse function Show that the ... |
7 Rates of Change 215 To find the distance the rock falls from time the end of three seconds, the rock has fallen 16 had only fallen to 144 feet at the end of one second, note that at feet, whereas it 144 16 128 feet So during this time interval the rock traveled 128 feet. The distance traveled by the rock during other... |
(x) change in x f(b) f(a) b a Example 2 Rate of Change of Volume A balloon is being filled with water. Its approximate volume in gallons is V x 1 2 x 3 55 where x is the radius of the balloon in inches. Find the average rate of change of the volume of the balloon as the radius increases from 5 to 10 inches. Solution ch... |
68Β° 64Β° 60Β° 56Β° 52Β° 48Β° 44Β° 40Β° 4 A.M. 8 12 Noon Time Figure 3.7-1 16 20 P.M. 24 218 Chapter 3 Functions and Graphs Solution a. The graph shows that the temperature at 4 a.m. is f f 4 and the 1 Thus, the average rate of change 58Β°. 12 2 40Β° temperature at noon is of temperature is 1 2 change in temperature change in t... |
x b f(b) f (a) b a slope of secant line joining (a, f (a)) and (b, f (b)) on the graph of f The Difference Quotient Average rates of change are often computed for very small intervals, such as the rate from 4 to 4.01 or from 4 to 4.001. Since and 4.001 4 0.001, both cases are essentially the same: computing the rate f... |
of β§β¨β© 3 V(x h) x h 55 2 1 and simplify. V x 1 2 V(x) β§β¨β© x 3 55 1 55 11 55 1 3x 2 3xh h2 55 1 55 x 3 3x 2h 3xh 2 h 3 x 3 h To find the average rate of change as x changes from 8 to let x 8 h 0.01. and 3x 2 3xh h2 55 So the average rate of change is 3 82 3 8 0.01 2 2 1 0.01 1 55 2 8.01 8 0.01, 3.495 β Exercises 3.7 1.... |
to 48 10 16 24 32 40 48 Month 6. A certain company has found that its sales are related to the amount of advertising it does in trade magazines. The graph in the figure shows the sales (in thousands of dollars) as a function of the amount of advertising (in number of magazine ad pages). Find the average rate of change... |
to x 3 1 2 222 Chapter 3 Functions and Graphs 13. x f 2 1 from 2x 3 2x 2 6x 5 x 1 to x 2 14. f x 2 1 from x 2 3 2x 4 x 3 to x 6 In Exercises 15β22, compute the difference quotient of the function. 15. 17. 19 16. 18. f f x x 1 1 2 2 7x 2 x 2 3x 1 160,000 8000t t2 20. V 22 21. A r 2 1 pr 2 23. Water is draining from a l... |
speed of car C from t 10. t 2 to c. Use secant lines and slopes to justify the statement βcar D traveled at a higher average speed than car C from t 10. t 4 to β e c n a t s i D 1200 1000 800 600 400 200 D C Car C Car D 2 4 6 8 10 12 Time 14 16 18 28. The figure below shows the profits earned by a certain company duri... |
........................ 142 Functions defined by equations.............. 144 Difference quotient........................ 144 Domain convention....................... 145 Piecewise-defined and greatest integer functions.............................. 146 Functions defined by graphs................ 150 Vertical Line Test... |
............. 169 Parent functions.......................... 172 Vertical shifts............................ 174 Horizontal shifts.......................... 175 Reflections across the x- and y-axis........... 177 Vertical and horizontal stretches and compressions.......................... 179 Combining transformations.... |
points and periodic orbits............. 202 Inverse relations and functions.............. 204 Graphs of inverse relations................. 206 One-to-one functions...................... 208 Horizontal Line Test....................... 208 Inverse functions......................... 210 Composition of inverse functions..... |
. is the graph of f shifted downward is the graph of f shifted c units to is the graph of f shifted c units to is the graph of f reflected across the is the graph of f reflected across the is the graph of f stretched or com- is the graph of f stretched or com- Horizontal Line Test: A function is one-to-one if and only ... |
dollars) spent on tickets for major concerts in selected years. [Source: Pollstar] s n o i l l i M 1500 1200 900 600 0 1990 1991 1992 1993 1994 1995 1996 a. What is the domain of the function? b. What is the approximate range of the function? c. Over what one-year interval is the rate of change the largest? In Exercis... |
Exercises 33β38, graph each function with its parent function on the same graph. 33. f x 1 2 2x 36. h x 2 1 x 3 4 34. h x 2 1 1 x 2 37. f x 1 2 x 2 3 35. x g 1 2 1.5 x 0 0 38. g x 1 2 23 2x In Exercises 39β42, list the transformations, in the order they should be performed on the graph of to produce a graph of the fun... |
. Plot the points 2, 1, f x 2 1 1, 3 x 1 x 0 0 51. h x 2 1 3x, 1, 3, 2, and 4, 1 on coordinate axes. 1 2 1 2 points 2 a. Suppose the points lie on the graph of an even function f. Plot the, 1 3, f 1 b. Suppose the points lie on the graph of an odd function g. Plot the points 3 22 1 3, g 2 3 4, g 4, f, and, and 4 4 0, g... |
Write the rule of a function g that gives the area of the spill at time t. c. What are the radius and area of the spill after 9 hours? d. When will the spill have an area of 100,000 square meters? Section 3.5.A 65. Find the first eight terms of the orbit of under the function 66. Describe the set of fixed points of th... |
1 1 2 2 4x 1 x 2 x 89. The graph of the function g in the figure consists of straight line segments. Chapter Review 233 Find an interval over which the average rate of change of g is 3 a. 0 b. d. Explain why the average rate of change of g is the same from c. 0.5 as it is from 2.5 to 0. 3 to 1 y 2 β2 β4 β2 x 2 4 6 8 9... |
0 Example 1 Instantaneous Velocity A ball is thrown straight up from a rooftop with an initial height of 160 feet and an initial velocity of 48 feet per second. The ball misses the rooftop on its way down and falls to the ground. Find the instantaneous velocity of the ball at seconds. t 2 Solution The height of the bal... |
Therefore, the instantaneous velocity is the remaining term of the difference quotient at t 2, In a similar fashion, the general expression of the difference quotient, becomes namely 16. 16h 32t 16h 48, 32t 48 Instantaneous rate of change expression when h is very small. Instantaneous rate of change can be found by us... |
time. Maximum height is given by s(1.5). 5 Figure 3.C-2 1.5 s 1 2 1 1 16 2 48 1.5 2 1 48 16 2.25 1 36 72 160 196 feet. 2 1.5 2 1.5 2 160 160 The graph of s and the tangent line to the curve at Figure 3.C-2. Notice that the tangent line is horizontal when are shown in t 1.5. t 1.5 β Writing the Equation of a Tangent Li... |
t seconds that is thrown straight up from a bridge at an initial height of 75 ft with an initial velocity of 20 ft/sec. Find the instantaneous velocity of the ball at seconds. t 2 2. Find when the instantaneous velocity of the ball in Exercise 1 is 0 feet per second, and interpret the result. 3. Write an equation for ... |
The Fundamental Theorem of Algebra Chapter Review can do calculus Optimization Applications Interdependence of Sections 4.3 4.1 > 4.2 > > > 4.4 4.5 > 4.6 P olynomial functions arise naturally in many applications. Many com- plicated functions can be approximated by polynomial functions or their quotients, rational fun... |
an1, p, a1, a0 where is a nonnegative integer. Definition of a Polynomial Function NOTE The term polynomial may refer to a polynomial expression or a polynomial function. The context should clarify the meaning. Review addition, subtraction, and multiplication of polynomials in the Algebra Review Appendix, if needed. P... |
x 1 3x 6 5 d Quotient d Dividend d Remainder 8x β CAUTION Synthetic division can be used only when the divisor is x c. Synthetic Division When the divisor is a first-degree polynomial such as there is a convenient shorthand method of division called synthetic division. The problem from Example 1 is reconsidered below, ... |
the divisor as To divide x 3 1 2 by x 3, x5 5x4 6x3 x2 4x 29 and perform synthetic division. 1 0 1 x4 2x3 x 7 29 21 The last row shows that the quotient is der is 8. and the remain- β Checking Polynomial Division Recall how to check a long division problem. When 4509 is divided by 31, the quotient is 145 and the remai... |
nomial and Rational Functions 2x2 1 Therefore, tor is the quotient, 1 is a factor of 3x 2. 2 2 1 6x3 4x2 3x 2, and the other fac- β Remainders When a polynomial x 3 x 5, or f x 1 2 is divided by a first-degree polynomial, such as the remainder is a constant. For example, when x3 2x2 4x 5 is divided by x 3, 2 f x 1 x2 x... |
f 1 2 2 Factor Theorem A polynomial function only if f(x) has a linear factor x a if and f(a) 0. The Factor Theorem states If a is a solution of f(x) 0, then x a is a factor of f(x). and If x a is a factor of f(x), then a is a solution of f(x) 0. You can see why the Factor Theorem is true by noting that the remainder ... |
function f β’ β’ x r x r is a solution, or root, of the equation f(x) 0 is a factor of the polynomial f(x) There is a one-to-one correspondence between the linear factors of of the graph of f. that have real coefficients and the x-intercepts f(x) The box above states that a zero, an x-intercept, a solution, and the valu... |
5 b 3 5 x 3 1 3 3b a 2a x 2 a 3b 5x 12 5 a 3x 2 21 x 3 x 3 1 1 2 21 2 β Example 8 A Polynomial with Specific Zeros Find three polynomials of different degrees that have 1, 2, 3, and zeros. 5 as Solution A polynomial that has 1, 2, 3, and x 3, 5 and conditions, such as x 5 x 1 2 5 x 2, as factors. Many polynomials sati... |
4x2 32x 5 5. 7. 21 x 23 2 B x 23 A B A 7 x2 5 x 15 8. x 1 k 2 1 (where k is a fixed positive integer) In Exercises 9β16, use synthetic division to find the quotient and remainder. 3x4 8x3 9x 5 9. 1 4x3 3x2 x 7 2 1 2x4 5x3 2x 8 3x3 2x2 8 1 5x4 3x2 4x 6 2 3x4 2x3 7x 4 10. 11. 12. 13. 14 In Exercises 17β22, state the quo... |
x5 5x4 5x3 25x2 6x 30 f(x) In Exercises 51β 54, each graph is of a polynomial function of degree 5 whose leading coefficient is 1, but the graph is not drawn to scale. Use the Factor Theorem to find the polynomial. Hint: What are the zeros of? What does the Factor Theorem tell you? f(x) 51. y In Exercises 31β40, find ... |
, 7, 4 56. n 3; zeros, 1, 1 57. n 6; zeros 1, 2, p 58. n 5; zero 2 59. Find a polynomial function f of degree 3 such that 17 10 f 1 2 and the zeros of f are 0, 5, and 8. x 2 1 60. Find a polynomial function g of degree 4 such that the zeros of g are 0, g 288. 3 1 2 1, 2, 3, and In Exercises 61 β 64, find a number k sat... |
. 5x 3, is the same as solving the Finding the real zeros of a polynomial The zero of a first-degree polynorelated polynomial equation, mial, such as can always be found by solving the equation 5x 3 0. Similarly, the zeros of any second-degree polynomial can be found by using the quadratic formula, as discussed in Sect... |
. 1, 1, 2, 2, 3, 3, 6, 6 Graph 1 the x-axis, such as necessary because only the x-intercepts are of interest. in a viewing window that includes all of these numbers on 7 y 7. A complete graph is not 7 x 7 and 2 The graph in Figure 4.2-1a shows that the only numbers in the list that could possibly be zeros are 3, 1 2, a... |
β§βͺβͺβͺβͺβ¨βͺβͺβͺβͺβ© 2x3 5x2 2x 2 2 Section 4.2 Real Zeros 253 2a x 1 2b1 2x2 4x 4 2 Therefore, 2x4 x3 17x2 4x 6 x 3 1 2a x 1 2b1 x 1 2b1 2x2 4x 4 x2 2x 2 2 2 2 1 x 3 2a The two remaining zeros of f are the solutions of can be found by using the quadratic formula 21 2 2 Β± 212 2 x2 2x 2 0, which Therefore, 2 212 2 f x 1 2 and h... |
4 7x3 13x2 3x 9 completely. Begin by finding as many rational zeros as possible. By the Rational Zero 10 Figure 4.2-2 Test, every rational zero is of the form where s Β± 1 Thus, the possible rational zeros are Β± 2. or r s, r Β±1, Β±3, or Β± 9 and Β± 1, Β± 3, Β± 9 The graph of f shows that the only possible zeros are easily ve... |
x4 x3 2x2 6x 21 2 x 7 3, the factor When, the is positive and quotient, is also positive because all its coefficients are positive. The x 7 3. x g remainder, 64, is also positive. Therefore, 1 x 7 3, g and so there are no zeros of g In particular, greater than 3. is positive whenever is never 0 when x 2 2 1 4 4 4 4 Fi... |
usually determine the number of real zeros the polynomial has, as shown in Example 4. The technique used in Example 4 to test possible lower and upper bounds works in the general case. 256 Chapter 4 Polynomial and Rational Functions Bounds Test Let f(x) be a polynomial with positive leading coefficient. β’ If d 77 0 di... |
1. Use the Rational Zero Test to find all the rational zeros of f. [Examples 1, 3, and 6] 2. Write f(x) as the product of linear factors, one for each rational zero, and another factor g(x). [Examples 2 and 3] 3. If g(x) has degree 2, find its zeros by factoring or by using the quadratic formula. [Example 2] 4. If g(x... |
13, rational zeros, 1 and 2, and two are irrational, as determined by using a zero finder. x 1.7913 and 2 β 258 Chapter 4 Polynomial and Rational Functions Exercises 4.2 When asked to find the zeros of a polynomial, find exact zeros whenever possible and approximate the other zeros. In Exercises 1 β 12, find all the ra... |
4 6x3 16. x5 2x4 2x3 3x 2 17. x5 4x4 8x3 14x2 15x 6 18. x5 4x3 x2 6x In Exercises 19β22, use the Bounds Test to find lower and upper bounds for the real zeros of the polynomial. In Exercises 23β36, find all real zeros of the polynomial. 23. 2x3 5x2 x 2 24. t4 t3 2t2 4t 8 25. 6x3 11x2 6x 1 26. z3 z2 2z 2 27. x4 x3 19x2 ... |
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