document5

200

ExercisesandProblemsforSection5.1

Chapter Five TRANSFORMATIONSOF FUNCTIONSAND THEIRGRAPHS

Exercises

1. Using Table 5.1, complete the tables for g, h, k, m,where:

(a) g(x) = f(x - 1)

(c) k(x) = f(x) + 3

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

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

Table5.1

~~

~~

~~~~~~

In Exercises 2-5, graph the transformations of f(x) in Fig-ure 5.7.

~lkx01 3 62 4 5

Figure5.7

2. y = f(x + 2)

4. y = f(x - 1) - 5

3. y = f(x) + 2

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

8. Match the graphs in (a)-(f) with the formula 11(i)-(vi).

(i) y = Ixl(iii) y = Ix- 1.21

(v) y = Ix+ 3.41

(a) y

(c) y

(e) y

(ii) y = Ixl- 1.2

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

(b) y

x x

(d) y

x x

(I) y

x x

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

(a) f(x) + 5?

(c) f(x - 3) - 2?(b) f(x + 5)?

10. The domain of the function g(x) is -2 < x < 7. Whatis the domain of g(x - 2)?

11. The range of the function R( s) is 100 ::;;R( s) ::;;200,Whatis therangeof R(s) - 150?

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

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 12. y = m(n) + 1thegraphsofh(x)andg(x)? 14 - ( ). y - m n - 3.7

(I

)X

(I)

X+4

(I

)X-2

7. Letf(x) = "3 ,g(x) ="3 ,andh(x) ="3 16. y = m(n) + vIf34 How do the graphs of g(x) and h(x) compare to the

graph of f(x)? 18. y = m(n + 3) + 7

13. y = m(n + 1)

15. y = m(n - 3.7)

17. y = m(n + 2v12)

19. y=m(n-17)-159

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

20. y=k(w)-3 21. y = k( w - 3)

Problems

5.1 VERTICALAND HORIZONTALSHIFTS 201

22. y=k(w)+1.8

24. y = k(w + 2.1) - 1.3

23. y = k(w + vg)

25. y = k(w - 1.5) - 0.9-----

f(x)26. (a) UsingTable5.2,evaluate

(i) f(x) for x = 6.

(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) - f(2) for x = 8.

(ix) g(x + 1) - g(x) for x = 1.

(b) Solve

(ii) f(x) = 574.(i) g(x) = 6.(iii) g(x) = 281.

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

to theequationX3+ X2+ x - 10 = 7;2 - 8x - 6.

Table5.2

27. The graph of g(x) contains the point (-2,5). Write aformula for a translation of 9 whose graph contains thepoint

(a) (-2,8) (b) (0,5)

28. (a) Let f(x) = Gr + 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.(d) Calculatef( 4) - f(2). Drawa verticallinesegment

on 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?

y

x

-rn=t~LJ-1jo

Figure5.8

29. Thefunction pet) gives the number of people in a certainpopulation in year t. Interpret in terms of population:

(a) pet) + 100 (b) pet + 100)

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

31. Graph f(x) = In(lx - 31)and g(x) = In(lxl). Find thevertical 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 toshow that the graphs are vertical shifts of one another.

Explain in words the effects of the transformations in Exer-cises 33-38 on the graph of q(z). Assume a, b are positiveconstants.

33. q(z) + 3

35. q(z + 4)

37. q(z + b)- a

34. q(z)-a

36. q(z - a)

38. q(z - 2b) + ab

39. Suppose Sed) gives the height of high tide in Seattle ona 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 foothigher 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 asthe previous day's high tide in Seattle.

x 0 1 2 3 4 5 6 7 8 9

f(x) -10 -7 4 29 74 145 248 389 574 809

g(x) -6 -7 6 33 74 129 198 281 378 489

202 Chapter Five TRANSFORMATIONSOFFUNCTIONSAND THEIRGRAPHS

40. Table 5.3 contains values of f(x). Each function in parts(a)-(c) is a translation of f(x). Find a possible for-mula for each of these functions in terms of f. For ex-ample, given the data in Table 5.4, you could say thatk(x) = f(x) + 1.

Table5.3

x 7

24.5I(x)

Table5.4

x 7

25.5k(x)

41. For t 2': 0, let H(t) = 68 + 93(0.91)t give the temper-ature of a cup of coffee in degrees Fahrenheit t minutesafter 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?

5.2 REFLECTIONSANDSYMMETRY...

42. At a jazz club, the cost of an evening is based on a covercharge of $20 plus a beverage charge of $7 per drink.

(a) Find a formula for t(x), the total cost for an eveningin which x drinks are consumed.

(b) If the price of the cover charge is raised by $5, ex-press the new total cost function, n(x), as a transfor-mation of tex).

(c) The -~ement increases the cover charge to $30,leaves the price of a drink at $7, but includes the firsttwo drinksfor free. For x 2':2, expressp(x), the newtotal cost, as a transformation oft(x).

7

22.5

43. A hot brick is removed from a kiln and set on the floorto cool. Let t be time in minutes after the brick was re-

moved. The difference, D(t), between the brick's tem-perature, initially 350°F, and room temperature, 70°F,decays exponentially over time at a rate of 3% perminute. The brick's temperature, H(t), is a transforma-tion of D(t). Find a formula for H(t). Compare thegraphs of D(t) and H(t), paying attention to the asymp-totes.

7

32 44. Suppose T(d) gives the average temperature in yourhometown on the dth day of last year (where d = 1 isJanuary 1st, and so on).7

30 (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 com-pare?

(e) Do you think the function T (d) + 365 has any prac-tical significance?Explain.

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

In Section 5.1 we saw that a horizontal shift of the gtaph of a function results from a change to theinput of the function. (Specifically,adding or subtracting a constant inside the function's parenthe-ses.) 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. Areflection about the x-axis corresponds to an outside change to the function's formula; a reflectionabout the y-axis and corresponds to an inside change.

(a) x-hex)

(b) -x-g(x)

(c)-x-i(xl


Exercises

5.2 REFLECTIONSAND SYMMETRY 209

1. The graph of y = f(x) contains the point (2, -3). Whatpoint must lie on the reflected graph if the graph is re-flected

(a) About the y-axis? (b) About the x-axis?

2. The graph of P = get) contains the point (-1, -5).

(a) If the graph has even symmetry, which other pointmust 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. IfH( -3) = 7, what is H(3)?

4. The range of Q(x) is -2 S Q(x) S 12. What is therange of -Q(x)?

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

6. If the graph of y = eX is reflected about the y-axis, what

is the formula for the resulting graph? Check by graphingboth functions together.

7. Complete the following tables using f(p) = p2 +2p-3,

and g(p) = f( -p), and h(p) = - f(p). Graph the threefunctions. Explain how the graphs of 9 and h are relatedto the graph of f. .

p

f(p)

3

p

g(p)

3

E-h(p)

Problems

8. Graphy = f(x) = 4x and y = f( -x) onthe samesetof axes.Howare thesegraphsrelated?Givean explicitformulafory = f( -x).

9. Graphy = g(x) = at and y = -g(x) on the sameset of axes. How are these graphs related? Give an ex-plicit formula for y = - 9(x).

Give a formula and graph for each of the transformations ofmen) = n2 - 4n + 5 in Exercises 10-13.

10. Y = m(-n)

12. y = -me -n)

11. y = -men)

13. y = -me -n) + 3

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

14. y = k(-w)

16. y=-k(-w)

15. y = -k(w)

17. y = -k(w - 2)

18. y = k( -w) + 4 19. y=-k(-w)-l

In Exercises 20-23, show that the function is even, odd, orneither.

20. f(x) = 7X2 - 2x + 1

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

21. f(x) = 4X7 - 3X5

23. f (x) = X5 + 3x3 - 2

24. (a) Graph the function obtained from f(x) = x3 by

first reflecting about the x-axis, then translating uptwo units. Write a formula for the resulting function.

(b) Graph the function obtained from f by first trans-lating 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 firstreflecting about the y-axis, then translating down

tp-ee units. Write a formula for the resulting func-tion.

(b) Graph the function obtained from 9 by first trans-lating down three units, then reflecting about the y-axis. 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 they-axis, what are the slope and intercepts of the resultingline?

27. Graph y = log(l/x) and y = log x on the same axes.How are the two graphs related? Use the properties oflogarithms to explain the relationship algebraically.

210 Chapter Five TRANSFORMATIONSOF FUNCTIONSAND THEIRGRAPHS

28. The function d(t) graphed in Figure 5.21 gives the wintertemperature in of at a high school, t hours after midnight.

(a) Describe in words the heating schedule for thisbuilding during the winter months.

(b) Graph c(t) = 142 - d(t).(c) Explain why c might describe the cooling schedule

for summer months.

temperature(OF)

68°d(t)

60° -

I

l t (hours)4 8 16 20 2412

Figure5.21

29. Using Figure 5.22, match the formulas (i)-(vi) with agraph from (a)-(f).

(i) y = f(-x)

(iii) y = f(-x) + 3(v) y = -f(-x)

(ii) y = - f(x)(iv) y = -f(x - 1)(vi) y = -2 - f(x)

-¥ f~X)Figure 5.22

(a)

*-X ~I ~y y

(c) y

(e)

Y (d)

4-x -~_x

Y m Y

f-x n~x

30. In Table 5.8, fill in as many y-values as you can if youknow that f is

(a) An even function (b) An odd function.

Table5.8

x 3

Y

31. Figure 5.23 shows the graph of a function f in the sec-ond quadrant. In each of the following cases, sketchy = f(x), given that f is symmetric about

(a) The y-axis. (b) The origin. (c) The line y = x.

Y

f(x) ~Figure5.23

32. For each table, decide whether the function could besymmetric about the y-axis, about the origin, or neither.

(a) 3

6

x

x f(x)

(d) 3

13

3

8.1

(b) x

g(x)

(c) - x

f(x) + g(x)


Solution To combine several transformations, always work from inside the parentheses outward as in Fig-ure 5.29. The graphs corresponding to each step are shown in Figure 5.30. Note that we did notneed a formula for f to graph g.

Step 1: horizontalshifllett 3 units

Step 2: vE\rticalcompression

j~

g(x) = -~f(x + 3) - 1

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

Step 4: verticalshift down 1 unit

Figure5.29


Exercises

Step1:y = f(x + 3)

~\\

-+\\ \

y

'15

Step2:1

Y = 2 f(x + 3)

III/"

I Original/ y=f(x)

//

/x

Step3:1

y = -- f(x + 3)2

4

-4

Figure 5.30: The graph of y = f (x) transformed in four steps intog(x) = -(1/2)f(x + 3) - 1

9. Using Table 5.14, create a table of values for1. Let y = f(x). Write a formula for the transformationwhich both increases the y-value by a factor of 10 andshifts the graph to the right by 2 units.

2. Thegraph of the function g(x) contains the point (5, ~).What point must be on the graph of y = 3g(x) + I?

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), -~f(x), and-5f(x) on the same axes.

4. f(x) = v'x

6. f(x) = eX

5. f(x) = -x2 + 7x

7. f(x) = lnx

8. Using Table 5.13, make tables for the following transfor-mations of f on an appropriate domain.

(a) ~f(x)(d) f(x - 2)

(b) -2f(x + 1) (c) f(x) + 5(e) f( -x) (f) - f(x)

Table5.13

x

J(x)

(a) f( -x) (b) - f(x) (c) 3f(x)

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

Table5.14

x 4

13J{x)

10. Figure 5.31 is a graph of y = x3/2. Match the followingfunctions with the graphs in Figure 5.32.

(a) y=x3/2-1 (b) y=(x-1)3/2

(c) y = 1 - x3/2 (d) y = ~X3/2

y

:vI i

y

2

(I) (II) (III)

1

t- x3

t2

t- x3

Figure 5.31 Figure 5.32

Without a calculator, graph the transfonnations in Exer- (a)cises 11-16. Label at least three points.

11. y = f(x + 3) if f(x) = Ix I12. y = f(x) + 3 if f(x) = Ixl13. y = -g(x) if g(x) = x2

14. y = g(-x) if g(x) = x2

15. y = 3h(x) if h(x) = 2'"16. y = 0.5h(x) if h(x) = 2'"

17. Using Figure 5.33, match the functions (i)-(v) with agraph (a)-(i).

(i) y = 2f(x)

(iii) y = -f(x) + 1

(v) Y = f(-x)

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

y

,/\ I ,/ y=f(x)

-r r--V--+- x-,

Figure5.33

Problems

5.3 VERTICALSTRETCHESAND COMPRESSIONS 217

y (b) y

~"~"(c) y

~x1 ~"

(d) y

(e) y y(I)

+" ~"(g) y

<VL ~"(h) y

(i) y

¥"

18. Describe the effect of the transfonnation 2f(x + 1) - 3on the graph of y = f(x).

19. The function s(t) gives the distance (miles) in tenns oftime (hours). If the average rate of change of s(t) on0 S t S 4 is 70 mph, what is the average rate of changeof ~s( t) on this interval?

In Problems 20-24, let f(t) = 1/(1 +X2). Graph the functiongiven, labeling intercepts and asymptotes.

20. y = f(t)

22. y = 0.5f(t)

24. y = f(t + 5) - 5

25. The number of gallons of paint, n = f (A), needed tocover a house is a function of the surface area, in ft2.Match each story to one expression.

(a) I figured out how many gallons I needed and thenbought two extra gallonsjust in case.

(b) I bought enough paint to cover my house twice.(c) I bought enough paint to cover my house and my

welcome sign, which measures 2 square feet.

21. y = f(t - 3)

23. y = -f(t)

(i) 2f(A) (ii) f(A + 2) (iii) f(A) + 2

26. The US population in millions is P(t) today and t is inyears. Match each statement (I)-(IV) with one of the for-mulas (a)-(h).

I. The population 10years before today.

II. Today's population plus 10million immigrants.

III. Ten percent of the population we have today.

N. The population after 100,000 people have emigrated.

(a) P(t) - 10 (b) P(t - 10) (c) O.lP(t)

(d) P(t) + 10 (e) P(t + 10) (f) P(t)/O.l

(g) P(t) + 0.1 (h) P(t) - 0.1

27. Let R = P( t) be the number of rabbits living in the na-tional park in month t. (See Example 5 on page 5.) Whatdo the following expressions represent?

(a) P(t + 1) (b) 2P(t)

218 ChapterFive TRANSFORMATIONSOF FUNCTIONSANDTHEIRGRAPHS

y28. Without a calculator, match each formula (a)-(e) with a

graph in Figure 5.34. There may be no answer or severalanswers.

(a) y=3.2x (b) y=5-x (c) y=-5x

(d) y = 2-Tx (e) y = I-or

y y

$x

x

y y

r xh(x)

x

Figure5.34

Graph the transformations of f in Problems 29-33 using Fig-ure 5.35. Label the points corresponding to A and B.

y10

B6

10

2x

-4

Figure 5.35

29. y = f(x - 3)

31. y=f(-x)/3

30. y=f(x)-3

32. y = -2f(x)

33. y = 5 - f(x + 5)

34. Using Figure 5.36, find formulas, in terms of f, for thehorizontal and vertical shifts of the graph of f in parts(a)--::(c).What is the equation of each asymptote?

5

3

(a) y

2

-1

(c) y

-3

35. Using Figure 5.37, find formulas, in terms of f, for thetransformations of f in parts (a)-(c).

y3 B

0-"I 2

Figure5.37

y y(a)

(b),~

h-~ x 21~-61V(c)y

l~x

1 rI I I I

-3 -1-t13 5 7

Figure5.36

y (b)


Exercises

5.4 HORIZONTALSTRETCHESAND COMPRESSIONS 223

1. The point (2,3) lies on the graph of g(x). What pointmust lie on the graph of g(2x)?

2. Describe the effect of the transformation 10f(fox) onthe graph of f(x).

3. Using Table 5.18, make a table of values for f( !x) foran appropriate domain.

Table5.18

x

f(x)

4. Fill in all the blanks in Table 5.19 for which you havesufficient information.

Table5.19

5. Graph m(x) = eX,n(x) = e2x,andp(x) = 2ex on thesame axes and describe how the graphs of n (x) and p (x)compare with that of m(x).

6. Graph y = h(3x) if hex) = 2x.

In Exercises 7-9, graph and label f(x), f(!x), and f( -3x)on the same axes between x = - 2 and x = 2.

7. f(x) = eX + X3 - 4X2

8. f(x) = eX+7 + (x - 4)3 - (x + 2?

9. f(x) = In(x4 + 3x2 + 4)

Problems

10. Using Figure 5.47, match each function to a graph (ifany) that represents it:

(i) y=f(2x) (ii) y=2f(2x) (iii) Y=f(!x)

yy = f(x)v x

Figure 5.47

(a)

(c)

(g)

11. For the function f (p) an input of 2 yields an output valueof 4. What value ofp would you use to have f(3p) = 4?

12. The domain of lex) is -12 :s; x :s; 12 and its range is0 :s; l (x) :s; 3. What are the domain and range of

(a) l(2;)? (b) l(!x)?

13. The point (a, b) lies on the graph of y = f(x). If thegraph is stretched away from the y-axis by a factor of d(where d > 1), and then translated upward by c units,what are the new coordinates for the point (a, b)?

x -3 -2 -1 0 1 2 3

f(x) -4 -1 2 3 0 -3 -6

f( x)

f(2x)

Y (b) Y

, ,,Y (d) Y

¥' ,(e) Y (f) Y

I tV "

,Y (h) Y

,0/1, ,(i) Y

-, LV +x

224 Chapter Five TRANSFORMATIONSOFFUNCTIONSAND THEIRGRAPHS

IIIIn Problems 14-15, graph the transformation of f, the func-tion in Figure 5.48.

14. y = -2f(x - 1) 15. y = f(x/2) - 1

16. Every day I take the same taxi over the same route fromhome to the train station. The trip is x miles, so the costfor the trip is f(x). Match each story in (a)-(d) to a func-tion in (i)-(iv) representing the amount paid to the driver.

(a) I received a raise yesterday, so today I gave mydriver a five dollar tip.

(b) I had a new driver today and he got lost. He drovefive extra miles and charged me for it.

(c) I haven't paid my driver all week. Today is Fridayand I'll pay what I owe for the week.

(d) The meter in the taxi went crazy and showed fivetimes the number of miles I actually traveled.

(i) 5f(x)

(iii) f(5x)

(ii) f(x) + 5

(iv) f(x + 5)

17. A companyprojectsa totalprofit,pet) dollars,in yeart. Explain the economic meaning of r(t) = 0.5P(t) andset) = P(0.5t).

18. Let A = f (r) be the area of a circle of radius r.

(a) Write a formula for fer).(b) Which expression represents the area of a circle

whose radius is increased by 1O%?Explain.

(i) O.IOf(r)(iv) f(Ur)

(ii) f(r+O.IO) (iii) f(O.IOr)(v) f(r)+O.IO

(c) By what percent does the area increase if the radiusis increased by 10%?

In Problems 19-20, state which graph represents

(a) f(x) (b) f(-2x) (c) f(-~x) (d) f(2x)

19. 20.

x

IV::

21. Find a formula for the function in Figure 5.50 as a trans-formation of the function f in Figure 5.49.

y3 B

~~ x

Figure5.49

~x -3Figure5.50

22. This problem investigates the effect of a horizontaIstretch on the zeros of a function.

(a) Graph f(x) = 4 - X2. Mark the zeros of f on thegraph.

(b) Graph and find a formula forg(x) = f(0.5x). Whatare the zeros of g(x)?

(c) Graph and find a formula for hex) = f(2x). Whatare the zeros of hex)?

(d) Without graphing, what are the zeros of f(1Ox)?

23. In Figure 5.51, the point e is labeled on the x-axis. Onthe y-axis, locate and label output values:

(a) gee) (b) 2g(e) (c) g(2e)

y

lL ~,(x)xc

Figure 5.51

y

21 II(x)

--I x-2 -1

I1 2

-1

-2

Figure5.48


Exercises

5.5THE FAMILYOF QUADRATICFUNCTIONS 231

yFor the quadratic functions in Exercises 1-2, state the coor-dinates of the vertex, the axis of symmetry, and whether theparabola opens upward or downward.

1. f(x) = 3(x -1)2 + 2

2. g(x) = -(x + 3? - 4

3. Sketch the quadratic functions given in standard form.Identify the values of the parameters a, b, and c. Labelthe zeros, axis of symmetry, vertex, and y-intercept.

(a) g(x)=x2+3 (b) f(x)= -2x2+4x+16 13.

4. Find the vertex and axis of symmetry of the graph ofv(t)= t2 + lIt - 4.

5. Find the vertex and axis of symmetry of the graph ofw(x) = -3X2 - 30x + 31.

6. Show that the function y = _X2 + 7x - 13 has no realzeros.

7. Find the value of k so that the graph of y = (x - 3)2 + kpasses through the point (6, 13).

8. The parabola y = ax2 + k has vertex (0, -2) and passesthrough the point (3,4). Find its equation.

In Exercises 9-14, find a formula for the parabola. -

9. 10. yy(4,7)

x

Problems

11. 12.y

x2~

-5

y 14.y (2,36)

For Exercises 15-18, convert the quadratic functions to ver-tex form by completing the square. Identify the vertex and theaxis of symmetry.

x

15. f(x) = X2 + 8x+ 3

16. g(x) = -2X2 + 12x + 4

17. Using the vertex form, find a formula for the parabolawith vertex (2,5) which passes through the point (1,2).

18. Using the factored form, find the formula for the parabolawhose zeros are x = -1 and x = 5, and which passesthrough the point (-2,6).

In Problems 19-24, find a formula for the quadratic functionwhose graph has the given properties.

19. A vertex at (4,2) and a y-intercept of y = 6.20. A vertexat (4,2) anda y-interceptof y = -4.21. A vertexat (4,2) andzerosat x = -3,11.22. A y-intercept of y = 7 and x-intercepts at x = 1,4.

23. A y-intercept of y = 7 and one zero at x = -2.

24. A vertex at (-7, -3) and contains the point (-3, -7).

25. Graph y = X2 - lOx + 25 and y = X2. Use a shifttr'l!1sformationto explain the relationship between thetwo graphs.

26. (a) Graph h(x) = -2x2 - 8x - 8.(b) Comparethegraphsof h(x) and f (x) = X2. How

are these two graphs related? Be specific.

27. Let f be a quadratic function whose graph is a concaveup parabola with a vertex at (1, -1), and a zero at theorigin.

(a) Graph y = f(x).

(b) Determine a formula for f(x).(c) Determine the range of f.

(d) Find any other zeros.


28. Let J(x) = X2andlet g(x) = (x - 3)2+ 2.(a) Give the formula for 9 in terms of J, and describe

the relationship between J and 9 in words.

(b) Is 9 a quadratic function? If so, find its standardformand the parametersa, b, and c.

(c) Graph g, labeling all important features.

29. If we know a quadratic function J has a zero at x = -1and vertex at (1,4), do we have enough information tofind a formula for this function? If your answer is yes,find it; if not, give your reasons.

30. Gwendolyn, a pleasant parabola, was taking a peacefulnap when her dream turned into a nightmare: she dreamtthat a low-flying pterodactyl was swooping toward her.Startled, she flipped over the horizontal axis, darted up(vertically) by three units, and to the left (horizontally)by two units. Finally she woke up and realized that her

equation was y = (x - 1? + 3. What was her equationbefore she had the bad dream?

31. A tomato is thrown vertically into the air at time t = O.Its height, d(t) (in feet), above the ground at time t (inseconds) is given by

d(t) = -16t2 + 48t.

(a) Graph d(t).(b) Find t when d(t) = O. What is happening to the

tomato the first time d(t) = O? The second time?(c) When does the tomato reach its maximum height?(d) What is the maximum height that the tomato

reaches?

32. An espressostandfindsthat its weeklyprofitis a func-tion of the price, x, it charges per cup. If x is in dollars,the weekly profit is P(x) = -2900X2 + 7250x - 2900dollars.

(a) Approximate the maximum profit and the price percup that produces that profit.

(b) Whichfunction,P(x-2) or P(x) -2, givesafunc-tion that has the same maximum profit? What priceper cup produces that maximum profit?

(c) Which function, P(x + 50) or P(x) + 50, gives afunctionwhere thepriceper cup that producesthemaximum profit remains unchanged? What is themaximum profit?

33. If you have a string of length 50 cm, what are the di-mensions of the rectangle of maximum area that you canenclose with your string? Explain your reasoning. Whatabouta stringof lengthk cm?

34. A footballplayerkicksaballatanangleof 37°abovethegroundwith an initial speedof 20 meters/second.Theheight, h, as a function of the horizontal distance trav-eled, d, is given by:

h = 0.75d - 0.0192d2.

(a) Graph the path the ball follows.(b) When the ball hits the ground, how far is it from the

spot where the football player kicked it?(c) What is the maximum height the ball reaches during

its flight?(d) What is the horizontal distance the ball has traveled

when it reaches its maximum height?3

35. A ballet dancerjumps in the air. The height, h( t), in feet,of the dancer at time t, in seconds since the start of the

jump, is given by4

h(t) = -16t2 + 16Tt,

where T is the total time in seconds that the ballet danceris in the air.

(a) Why does this model apply only for 0 ::;t ::;T?(b) When, in terms of T, does the maximum height of

the jump occur?(c) Show that the time, T, that the dancer is in the air is

related to H, the maximum height of the jump, bythe equation

H = 4T2.

3Adapted from R. Halliday, D. Resnick, and K. Krane, Physics. (New York: Wiley, 1992), p.58.4K. Laws, The Physics afDance. (Schinner, 1984).


Exercises

8.3COMBINATIONSOF FUNCTIONS 379

In Exercises 1-6, find the following functions. 12. k(x) = m(x) - n(x) - o(x)

(a) f(x)+g(x)(e) f(x)g(x)

(b) f(x) - g(x)(d) f(x)/g(x)

1. f(x) = x + 1 g(x) = 3X2

2. f(x) = X2+ 4 g(x) = x + 2

3. f(x) = x + 5 g(x) = x - 5

4. f(X)=X2+4 g(x)=x2+2

5. f(X)=X3 g(X)=X2

6. f (x) = -IX g (x) = X2 + 2

In Exercises 13-16, let u( x) = eX and v (x) = 2x + 1. Finda simplified formula for the function.

13. f(x) = u(x)v(x)

15. h(x) = (v(U(X)))2

14. g(x) = U(X)2+V(X)2

16. k(x) = v(u(x?)

Find formulas for the functions in Exercises 17-22. Let

f(x) = sinx and g(x) = x2.In Exercises 7-12, find a simplified formula for the function.Let m(x) = 3X2 - x, n(x) = 2x, and o(x) = VX + 2.

17. f(x) + g(x) 18. g(x)f(x)

7. f(x) = m(x) + n(x)

9. h(x) = n(x)o(x)

8. g(x) = (O(X))2

10. i(x) = m(o(x))n(x)

11. j(x) = (m(x))/n(x)

Problems

19. f(x)/g(x)

21. g(f(x))

20. f(g(x))

22. 1 - (f(X))2

23. (a) Onthesamesetofaxes,graphf(x) = (X-4)2_2and g(x) = -(x - 2)2+ 8.

(b) Make a table of values for f and g for x =0,1,2, ,.,,6.

(e) Make a table of values for y = f(x) - g(x) forx=0,1,2,...,6.

(d) On your graph, sketch the vertical line segment oflength f (x) - g (x) for each integer value of x from0 to 6. Check that the segment lengths agree with thevalues from part (c).

(e) Plot the values from your table for the function

y = f(x) - g(x) on your graph.(f) Simplify theformulas for f(x) and g(x) in part (a).

Find a formula for y = f(x) - g(x).(g) Use part (f) to graph y = f(x) - g(x) on the same

axes as f and g. Does the graph pass through thepoints you plotted in part (e)?

24. Figure 8.26 shows a weight attached to the end of a springwhich is hanging from the ceiling. The weight is pulleddown from the ceiling and then released. The weight os-cillates up and down, but over time, friction decreasesthe magnitude of the vertical oscillations. Which of thefollowing functions could describe the distance of the

weight from the ceiling, d, as a function of time, t?

(i) d = 2 + cas t (ii) d = 2 + e-t cas t

(iii) d = 2 + cos(et) (iv) d = 2 + ecost

Figure8.26

25. Table 8.25 gives the upper household income limits forthe tenth and ninety-fifth percentiles t years after 1993.4For instance, Ho(5) = 9,700 tells us that in 1998 themaximum income for a household in the poorest 10% of

all households was $9,700. Let f(t) = P95(t) - PlO(t)andg(t) = P95(t)/Ho(t).

(a) Maketablesof valuesfor f andg.

4US Census Bureau, The Changing Shape of the Nation's Income Distribution, accessed December 29, 2005, atwww.census.gov/prod/2000pubs/p60-204.pdf.

380 ChapterEight COMPOSITIONS,INVERSES, ANDCOMBINATIONSOF FUNCTIONS

(b) Describe in words what f and 9 tell you about 29. Use Figure 8.29 to graph hex) = g(x) - f(x). On thehousehold income. graph of hex), label the points whose x-coordinates are

x = a, x = b, and x = c. Label the y-intercept.Table8.25

26. Use Table 8.26 to make tables of values for x = -1, 0,

1, 2, 3, 4 for the following functions.

(a) hex) = f(x) + g(x) (b) j(x) = 2f(x)

(c) k(x) = (g(x)? (d) m(x) = g(x)/f(x)

Table8.26

27. Use Figure 8.27 to graph the following functions.

(a) y = g(x) - 3 (b) y = g(x) + x

Y g(x)1

x

Figure8.27

28. Graph hex) = f(x) + g(x) usingFigure8.28.

~8 I

1 9x

Figure8.28

g(x)x

Figure8.29

30. (a) Find possible formulas for the functions in Fig-ure 8.30.

(b) Let hex) = f(x). g(x). Graph f(x), g(x) and hex)on the same set of axes.

g(x)X

Figure8.30

31. Use Figure 8.31 to graph c(x) = a(x) .b(x). [Hint:Thereis not enough information to determine formulas for aand b but you can use the method of Problem 30.]

a(x)Y

b(x)x

Figure8.31

t (yrs) 0 1 2

PlO(t)($) 8670 8830 9279

P95(t)($) 118,036 120,788 120,860

t (yrs) 3 4 5

PlO(t) ($) 9256 9359 9700

P95(t) ($) 124,187 128,521 132,199

x -1 0 1 2 3 4

I(x) -4 -1 2 5 8 11

g(x) 4 1 0 1 4 9

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