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Chapter Five TRANSFORMATIONS FUNCTIONS OF AND THEIRGRAPHS

Exercises andProblems Section for 5.1Exercises 1. Using Table 5.1, complete the tables for g, h, k, m, where: (a) (c) g(x) = f(x - 1) k(x) = f(x) + 3 8. Match the graphs in (a)-(f) with the formula 11 (i)-(vi).

(b) h(x) = f(x + 1) (d) m(x)=f(x-l)+3

= Ixl (iii) y = Ix- 1.21 (v) y = Ix+ 3.41(i) y (a)y

(ii) (iv)

y

= Ixl-

1.2

y = Ixl+ 2.5 (vi) y = Ix - 31+ 2.7 y

Explain how the graph of each function relates to the graph of f(x).

(b)

Table 5.1

~ ~ ~ ~ ~ ~ ~ ~ ~ ~In Exercises 2-5, graph the transformations of f(x) in Figure 5.7.(c) y

x

x

(d)

y

x

x

(e)

y

(I)

y

x

x

9. The graph of f(x) contains the point (3, -4). What point must be on the graph of

~lkx0 1 2 3

(a)

f(x) + 5?-

(b)

f(x + 5)?

(c) f(x4 5 6

3) - 2?

Figure5.7

10. The domain of the function g(x) is -2 < x < 7. What is the domain of g(x - 2)? 11. The range of the function R( s) is 100 ::;;R( s) ::;;200,

2. y 4. y

= f(x + 2) = f(x- 1) - 5

3. y = f(x) + 2

Whatis therangeof R( s)

-

150?

5. y = f(x + 6) - 4

Write a formula and graph the transformations of m( n) ~n2 in Exercises 12-19. 12. y

=

6. Let f(x) = 4x, g(x) = 4x + 2, and h(x) = 4x - 3. What is the relationship between the graph of f(x) and

= m(n) + 1

13. y

= m(n + 1)-

thegraphsofh(x)andg(x)?7. Letf(x) =4

("3) ,g(x) ="3 ) (

I

X

I

X+4

,andh(x) ="3

()

I

X-2

14. y - m (n ) - 3.7 16. y = m(n) + vIf318. y

15. y = m(n

3.7)

17. y = m(n + 2v12)19. y=m(n-17)-159

How do the graphs of g(x)

and h(x)

compare to the

graph of f(x)?

= m(n + 3) + 7

5.1 VERTICALAND HORIZONTAL SHIFTS

201

Write a formula and graph the transformations of k (w) = 3w in Exercises 20-25. 20. y=k(w)-3 21. y = k( w - 3)

22. y=k(w)+1.8 24. y = k(w + 2.1)-

23. y 1.3 25. y

= k(w + vg) = k(w-

1.5)

-

0.9 -----

Problems26. (a) Using Table5.2,evaluate (i) f(x) for x = 6.y f(x)

(ii) f(5) - 3. (iii) f(5 - 3).(iv) g(x) + 6 for x = 2. (v) g(x + 6) for x = 2. (vi) 3g(x)forx=0. (vii) f(3x) for x = 2.(viii) f(x)-

x

-rn=t~LJ-1jo8.

f(2) for x

=

Figure5.8

(ix) g(x + 1) - g(x) for x = 1. (b) Solve (i) g(x) = 6. (iii) g(x) = 281. (ii) f(x) = 574.

29. Thefunction pet) gives the number of people in a certain population in year t. Interpret in terms of population: (a) pet) + 100 (b) pet + 100)

(c) The values in the table were obtained using the formulas f(x) = X3 + X2 + x - 10 and g(x) = 7X2 - 8x - 6. Use the table to find two solutions

30. Describe a series of shifts which translates the graph of y = (x + 3)3 -1 onto the graph ofy = X3.

to theequationX3+ X2+ xTable5.2x f(x) g(x) 0 -10 -6 -7 -7 1 2 4 6 3 29 33 4 74 74 5 145 129

-

10 = 7;2 - 8x - 6.

31. Graph f(x) = In(lx - 31) and g(x) = In(lxl). Find the vertical asymptotes of both functions. 32. Graph y = log x, y = 10g(lOx), and y = 10g(1O0x). How do the graphs compare? Use a property of logs to show that the graphs are vertical shifts of one another. Explain in words the effects of the transformations in Exercises 33-38 on the graph of q(z). Assume a, b are positive constants. 33. q(z) + 3 35. q(z + 4) 34. q(z)-a 36. q(z - a) 38. q(z - 2b) + ab

6 248 198

7 389 281

8 574 378

9 809 489

27. The graph of g(x) contains the point (-2,5). Write a formula for a translation of 9 whose graph contains the point (a) (-2,8) (b) (0,5)

37. q(z + b) - a

28. (a) Let f(x) = + 2. Calculate f( -6). (b) Solvef(x) = -6. (c) Find points that correspond to parts (a) and (b) on the graph of f(x) in Figure 5.8.

Gr

39. Suppose Sed) gives the height of high tide in Seattle on a specific day, d, of the year. Use shifts of the function Sed) to find formulas for each of the following functions: (a) T(d), the height of high tide in Tacoma on day d, given that high tide in Tacoma is always one foot higher than high tide in Seattle. (b) P(d), the height of high tide in Portland on day d, given that high tide in Portland is the same height as the previous day's high tide in Seattle.

(d) Calculatef( 4) - f(2). Drawa verticallinesegmenton the y-axis that illustrates this calculation. (e) If a = -2, compute f(a + 4) and f(a) + 4. it) In part (e), what x-value corresponds to f(a + 4)? To f(a) + 4?

202

Chapter Five TRANSFORMATIONS FUNCTIONSAND THEIRGRAPHS OF

40. Table 5.3 contains values of f(x). Each function in parts (a)-(c) is a translation of f(x). Find a possible formula for each of these functions in terms of f. For example, given the data in Table 5.4, you could say that k(x) = f(x) + 1. Table5.3 xI(x)

42. At a jazz club, the cost of an evening is based on a cover charge of $20 plus a beverage charge of $7 per drink. (a) Find a formula for t(x), the total cost for an evening in which x drinks are consumed. (b) If the price of the cover charge is raised by $5, express the new total cost function, n(x), as a transformation of tex). (c) The -~ement increases the cover charge to $30, leaves the price of a drink at $7, but includes the first two drinks for free. For x 2':2, expressp(x), the new total cost, as a transformation oft(x). 43. A hot brick is removed from a kiln and set on the floor to cool. Let t be time in minutes after the brick was removed. The difference, D(t), between the brick's temperature, initially 350F, and room temperature, 70F, decays exponentially over time at a rate of 3% per minute. The brick's temperature, H(t), is a transformation of D(t). Find a formula for H(t). Compare the graphs of D(t) and H(t), paying attention to the asymptotes. 44. Suppose T(d) gives the average temperature in your hometown on the dth day of last year (where d = 1 is January 1st, and so on). (a) Graph T(d) for 1 ::; d ::; 365. (b) Give a possible value for each of the following: T(6); T(100); T(215); T(371). (c) What is the relationship between T(d) and T(d + 365)? Explain. (d) If you were to graph wed) = T(d + 365) on the same axes as T(d), how would the two graphs compare? (e) Do you think the function T (d) + 365 has any practical significance? Explain. 45. Let f(x) find h.

7 24.5

Table5.4x k(x) 7 25.5

(a)

x hex)

7 22.5

(b)

-x g(x)

7 327 30

(c)

-x i(xl

41. For t 2': 0, let H(t) = 68 + 93(0.91)t give the temperature of a cup of coffee in degrees Fahrenheit t minutes after it is brought to class. (a) Find formulas for H(t + 15) and H(t) + 15. (b) Graph H(t), H(t + 15), and H(t) + 15. (c) Describe in practical terms a situation modeled by the function H(t + 15). What about H(t) + 15? (d) Which function, H(t+15) or H(t)+15, approaches the same final temperature as the function H (t)? What is that temperature?

= eXand g(x) = 5ex. If g(x) = f(x - h),

5.2

REFLECTIONS ANDSYMMETRY ...In Section 5.1 we saw that a horizontal shift of the gtaph of a function results from a change to the input of the function. (Specifically, adding or subtracting a constant inside the function's parentheses.) A vertical shift corresponds to an outside change. In this section we consider the effect of reflecting a function's graph about the x or y-axis. A reflection about the x-axis corresponds to an outside change to the function's formula; a reflection about the y-axis and corresponds to an inside change.

5.2 REFLECTIONS AND SYMMETRY

209

Exercises andProblems Section5.2 for Exercises1. The graph of y = f(x) contains the point (2, -3). Whatpoint must lie on the reflected graph if the graph is reflected (a) About the y-axis? (b) About the x-axis?

8. Graphy = f(x) = 4x and y = f( -x) on the sameset of axes.Howare thesegraphsrelated?Givean explicit formulafory = f( -x). 9. Graphy

2. The graph of P

= get)

contains the point (-1, -5).

(a) If the graph has even symmetry, which other point must lie on the graph? (b) What point must lie on the graph of -get)? 3. The graph of H (x) is symmetric about the origin. If H( -3) = 7, what is H(3)?4. The range of Q(x) is -2 range of -Q(x)? S Q(x) S 12. What is the

= g(x) = and y = -g(x) on the same set of axes. How are these graphs related? Give an explicit formula for y = - 9(x).= n2- 4n + 5 in Exercises 10-13.

at

Give a formula and graph for each of the transformations ofmen)

10. Y = m(-n)

11. y = -men) 13. y = -me -n) + 3

12. y

= -me -n)

5. If the graph of y = eX is reflected about the x-axis, what is the formula for the resulting graph? Check by graphing both functions together.

6. If the graph of y

about the y-axis, what is the formula for the resulting graph? Check by graphing both functions together.

= eX is reflected

Give a formula and graph for each of the transformations of k(w) = 3w in Exercises 14-19. 14. y

7. Complete the following tables using f(p)

= p2

= k(-w)

15. y = -k(w) 17. y = -k(w - 2) 19. y=-k(-w)-l

+2p-3,

and g(p) = f( -p), and h(p) = - f(p). Graph the three functions. Explain how the graphs of 9 and h are related to the graph of f..

16. y=-k(-w)

18. y = k( -w) + 4

p f(p) p g(p)

3

3

In Exercises 20-23, show that the function is even, odd, or neither. 20. f(x) = 7X2 - 2x + 1 21. f(x)23. f (x)

= 4X7 -

3X5

Eh(p)

22. f(x) = 8x6 + 12x2

= X5

+ 3x3 - 2

Problems24. (a) Graph the function obtained from f(x)

=

x3 by

first reflecting about the x-axis, then translating up two units. Write a formula for the resulting function. (b) Graph the function obtained from f by first translating up two units, then reflecting about the x-axis. Write a formula for the resulting function. (c) Are the functions in parts (a) and (b) the same? 25. (a) Graph thefunction obtained from g(x) = 2x by first reflecting about the y-axis, then translating down tp-ee units. Write a formula for the resulting function.

(b) Graph the function obtained from 9 by first translating down three units, then reflecting about the yaxis. Write a formula for the resulting function. (c) Are the functions in parts (a) and (b) the same? 26. If the graph of a line y = b + mx is reflected about the y-axis, what are the slope and intercepts of the resulting line? 27. Graph y = log(l/x) and y = log x on the same axes. How are the two graphs related? Use the properties of logarithms to explain the relationship algebraically../

210

Chapter Five TRANSFORMATIONS FUNCTIONSAND THEIRGRAPHS OF

28. The function d(t) graphed in Figure 5.21 gives the winter temperature in of at a high school, t hours after midnight. (a) Describe in words the heating schedule for this building during the winter months.(b) Graph c(t)

30. In Table 5.8, fill in as many y-values as you can if you know that f is (a) An even function (b) An odd function.

= 142

- d(t).

(c) Explain why c might describe the cooling schedule for summer months.temperatureOF) ( 68

Table 5.8x Y 3

d(t)

60

I

l4 8 12 16 20 24

t (hours)

31. Figure 5.23 shows the graph of a function f in the second quadrant. In each of the following cases, sketch y = f(x), given that f is symmetric about (a) The y-axis. (b) The origin. (c) The line y

Figure5.21 29. Using Figure 5.22, match the formulas (i)-(vi) with a graph from (a)-(f). (i) y = f(-x) (iii) y = f(-x) (v) y = -f(-x)

= x.

Y

(ii) y +3(iv) y(vi) y

=-

f(x)- 1)f(x) f(x)

= -f(x = -2 -

Figure5.23

~3 6 3 8.1

-Figure 5.22

f~X)32. For each table, decide whether the function could be symmetric about the y-axis, about the origin, or neither.y

(a)

y

*-X(c) Y

~I(d)

~y -~_x Y

(a)x

x f(x)

(b)

x g(x)

(c) 4-x (e) Y m

x f(x) + g(x)

(d)

3 13

f-x

n~x

216

Chapter Five TRANSFORMATIONS FUNCTIONSAND THEIRGRAPHS OF

Solution

To combine several transformations, always work from inside the parentheses outward as in Figure 5.29. The graphs corresponding to each step are shown in Figure 5.30. Note that we did not need a formula for f to graph g.Step 1: horizontal shifllett 3 unitsStep 2: vE\rtical

Step = f(x + 3) 1:y

y

~Step2: 1 Y = 2 f(x

'15

\

compression~

\\ /

j

+ 3) -+\\

/ //

I

/"

I I I

Original

y=f(x)x

g(x) = -~f(x

+ 3) - 1Step3:

41 y = -- 2 f(x + 3) -4

""" 3,",1., "",,",,0across the x-axis

1Step 4: vertical shift down 1 unit

Figure5.29

Figure 5.30: The graph of y =g(x)

f (x) transformed in four steps into+ 3)-

= -(1/2)f(x

1

Exercises andProblems Section5.3 forExercises1. Let y = f(x). Write a formula for the transformation which both increases the y-value by a factor of 10 and shifts the graph to the right by 2 units. 9. Using Table 5.14, create a table of values for(a)

f( -x)

(b)

- f(x)

(c)

3f(x)

2. Thegraph of the function g(x) contains the point (5, ~).What point must be on the graph of y

= 3g( x) + I?

(d) Which of these tables framparts (a), (b), and (c) represents an even function?

3. The range of the function C (x) is -1 S; C (x) S; 1. What is the range of O.25C(x)? In Exercises 4-7, graph and label f(x), 4f(x),-5f(x) on the same axes.

Table5.14x J{x) 4 13

-~f(x),

and

4. f(x) = v'x 6. f(x)= eX

5. f(x)

= -x2 lnx

+ 7x

10. Figure 5.31 is a graph of y = x3/2. Match the following functions with the graphs in Figure 5.32. (a) (c)y

7. f(x) =

y=x3/2-1 y = 1 - x3/2

(b) y=(x-1)3/2 (d) y = ~X3/2y 2 1 t- x 3 (I) (II) (III)

8. Using Table 5.13, make tables for the following transformations of f on an appropriate domain. (a) (d) ~f(x) f(x - 2) Table5.13x J(x)

(b) -2f(x (e) f( -x)

+ 1) (c) f(x) + 5 (f) - f(x)

:vI

i

t 2

t- x 3

Figure 5.31

Figure 5.32

5.3 VERTICALSTRETCHESAND COMPRESSIONS

217

Without a calculator, graph the transfonnations cises 11-16. Label at least three points. 11. y

in Exer- (a)

y

(b)

y

12. y 13. y 14. y15. y

= f(x + 3) if f(x) = Ix I = f(x) + 3 if f(x) = Ixl = -g(x) if g(x) = x2

~"(c) y(d)

~"y

= g(-x)= 3h(x)

if g(x)if h(x)

= x2= 2'"(e)

~x

1y(I)

~"y

16. y = 0.5h(x) if h(x) = 2'"17. Using Figure 5.33, match the functions (i)-(v) with a graph (a)-(i). (i) y(iii)

y

(v)

= 2f(x) y = -f(x) Y = f(-x)

+ 1

(ii) y=~f(x) (iv) y = f(x + 2) + 1

(g)

+"

~"(h) y

y

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