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Unformatted text preview: 190 CHAPTER 2 Functions 1~1€l W Suppose the graph of f is given. Describe how the graph
of each function can he obtained from the graph of f. L(my=fu)w5 m)y=f©“5} 2. (a)ymf{x+7) (b) y=f(x)+7
3.(a)ymf(x+%) (b) y=f{x)+§ 4. (a) y 2: Mﬂx) (b) y 2 fWC) 5. (a) y 3 2f(x) (b) m W305) 6. (a) y = —f(x) + 5 (b) y m 3W) — 5
7.(a)y:f(x—4}'+% (b}y*f{x+4)%
swmy=wu+mu2 acyzﬁu—m+2
9. (a) y = mac)  (b) y = 16x) 10. (a) y = —f{2x) (b) y = fax) “ 1 1116 E The graphs of 3° and g are given. Find a formula for the
function g. a)y=%
“(CHWJ $9. [The graph
" functions. 17—18 H The graph of y 3 ﬁx) is given. Match each equation j
with its graph. ' 17. (a) y 3 ﬁx w 4)
(c) y = Zﬂx —i~ 6} (b) y 3 f{x)‘+ 3
(d) y E “fwd ® +"+i ! m t (x) (b) y : *f(x + 4)
(d) y = f{"x) a graph of f is given. Sketch the graphs of the following
ﬁctions. is graph of g is given. Sketch tho graphs of the foliowing
notions. to the graph off to sketch the granhs of the foltowing
actions. .. W _1
x (mymxwi l
x"3 = (iv)ym1~t~ SECTIGN 2.4 Transformations of Functions 191 22. (a) Sketch the graph of g(x) = “3/ch by plotting points. (b) Use the graph of g to sketch the graphs of the following
functions. (t)y=\3/£T“§
(iii)y=1\3/JE (it) ymx3/x+2+2
(iv) y22NZ/E 23—26 3 Explain how the graph of g is obtained from the graph
of f. 23. (a) f(x) = x2, g(x) : (x + 2}2
(5) fix} 5 x2: 9'06} m X: "i" 2
24. (a) ﬁx) w x3, g{x) = (A: ~ £83
(13) ﬁx) 2 x3, g(x) = x3 — 4
25. (a) ﬁx) = V22, 9(x) : 2%?
(is) ax) = Vi, an at “2
26 (a) fix IXI, 90?) z 3i?” +1
{b} ﬁx) L" WI» 90‘) = “Hi 1 3E 27—32 E A function f is given, and the indicated transforma—
tions are‘apolied to its graph (in the given order). Write the
equation for the ﬁnal transfonned graph. 27. f(x) = x2; shift upward 3 units and shift 2 units to the
right 28. ﬁx) m“ x3; shift downward 1 unit and shift 4 units to
the 26ft 29. ﬁx) m V}; shift 3 units to the left, stretch verticatty by a
factor of 5, and reﬂect in the x~axis 39. f(x) =_ 6%; reﬂect in the y—axis, shrink verticaliy by a fac—
tor of and shift upward i unit 31. f(x) 2 [x l ; shift to the righté unit, shrink verticaily by a
factor of 0.1, and shift downward 2 units 32. ﬁx) a Ix i ; shift to the ieft 1 unit, stretch vertically by a
factor of 3, and shift upward 10 units 33—48 3 Sketch the graph of the fonction, not by plotting
points,b12tby starting with the graph of a standard function and
applying transformations. 33. ﬁx) a (x»~~2)2 34. ﬁx) =(x+7)2
35. f(x)m m(xi1)2 36. _f(;c)m1wx2
37. ﬁx) =x3+2 38. ﬁx) m "x3
39.y=t+\/;Tc 40.y=2\/£Tt
41.ym% x+4m3 42.y=3_2(x1)2
43.ym5+{x+3)2 44.y:%x31
45.y=lxiI 46.y=ix1 47.y=1x+2i+2 48.y=2—xl ea
BE 192 CHAPTER 2 Funafions 4952 e Graph the functions on the same screen using the SS.
given viewing rectangle. How is each graph reiateti to the graph
in part (a)?
49. Viewing rectangle {—8, 8] by {"2, 8]
(a)y=\i€c (b)y=\/“x+5
(c)y:2\“/£"3¥"§ (d)y=4+2m 56. Viewing rectangle [w 8, 8} by [w 6, 6] (ﬂymixl (blyaﬂxl
(c) y=“3lx! (diymm3ixm5i
. . 57—58 m Use the graph of ﬁx) : described on pages
. " , — , 4 . . . 51 Viewmg ricwﬂglﬁ 4 63b“ 4 I} 6 162—163 to graph the indicated fonctxon. (£1)ny (b)y='3x (c) y 3 Mix“ {6) y = ~i(x — 4f 57 y e ilsz 58 y = iixl
52’ Viﬁwgng refmngiehﬁ’ GJbﬂmdi’ 4] 1 {ﬁg 59. If ﬁx) = V 2):  x2, graph the following functions inthﬁ (a) E. W (b) : __ viewing rectangiefmi S] by {"4, 4]. How is each graph 5" y V32 y Vx + 3 iated to the graphin past (a)?
1 ___1___  3 (a) y = f(x) 53) y 2 far) (0) y m fiixi i“) “W (“Nix/m 53. The graph of g is given. Use it to graph each of the
foilowing functions. (a) y = we o) y m nix) ﬁg 60. If ﬂx} m V 2x  x2, graph the following functions in $11"
viewing rectangle [5, 5} by [4, 4]. How is each graph
reiated to the gragh in part (a)?
(a) y 2 ﬁx) 0)) y m ﬁrst) (c) y e —f(
(d) y = f{2x} (e) y '—"—"’ f(—%x) 61w68 m Determine whether the function f is even, odd, or
neither. If f isieven or odd, use symmetry to sketch its graph. 6i. f(x} = )6"2 62. f(x) = x'3
63. ﬁx) m x2 + x 64. ﬂx) m x4 — 4x7”
65. ﬁx) = x3 — x 66. ﬁx) 2 3x3 + 2x2 + 1
54. The graph of h is given. Use it to graph each of the
following functions.  57‘ ﬁx) = 1 .— ﬁ 63_ ﬁx} 2 x 4;“ "1m
x (a) y = hex) (b) y w hex) 69. The graphs ofﬂx) = x2 —— ti and g(x} = 1x2  41
are shown. ExpEain how the graph of g is obtained from
the graph of f. 55—56 The gragh of a function éeﬁned for x a 0 is given.
Complete the graph for x < O to make (a) an even function
(h) an odd function e grape of f(x} =" x4  4x2 is shown. Use this graph to
' sketch the graph of g(x) 5 ix” m 4x2 \: ‘1 re“ bed'on pages ' 4xmx2 31 we
3
'3
V
ll
>1
w
A
a.
V
‘3‘
M
v
li
V g functions in
v is each graph (‘3) y m "f( ‘_ ates Growth The annual sales of a certain company
' an be modeleé by the function ﬁr) = 4 + 0.03:2, where represents years since 1990 and f(r) is measured in millions of dollars. Ia) What shifting and shrinking operations must be per
fortneti on the function y m r2 to obtain the function
y e ftr)?
) Suppose you want t to represent years since 2000 in
' stead of 1990. What transformation wouid you have to 5 even, out}, or
ketch its graph the new function y m g(r) that results from this trans
formation. apply to the function y w ﬁt) to accomplish this? Write SECTlON 2.5 Quadratic Functions; Maximo end Minima 193 '74. Changing Temperature Scaies The temperature on a certain afternoon is modeled by the function
60‘) m it? 43 2 where t represents hours after 12 noon (0 E r S 6). and C
is measured in °C. (a) What shifting and shrinking operations must be per
formed on the function y x r2 to obtain the function
y = CO)? (13) Suppose you want to measure the temperature in “F
instead. Mat transformation would you have to
apply to the function y m C(13) to accomplish this?
(Use the fact that the relationship between Celsius and
Fahrenheit degrees is given by F 3 2C ~l~ 32.)W1ite
the new function y m F{t) that results from this
transformation. Discovery e Discussion 75. 76. 77. Sums of Even and Odd Functions If f and g are both
even functions, is f + g necessarily even? If both are odd, is
their sum necessariiy odd? What can you say about the sum
if one is odd and one is even? In each case, prove your
answer. Products of Evan and Odd Functions Answer the same
questions as in Exercise 75, except this time consider the
product of f and 9 instead of the sum. Evan and Odd Power Functions What must be true
about the integer n if the function f(x) m x" is an even function? If it is an odd function? Why do you
think the names “even” and “odd” were chosen for these
function properties? sunerreas; "anagram an new ' A maximum or minimum value of a function is the largest or smallest value of the
function on an interval. For a function that represents the proﬁt in a business, we
would be interested in the maximum value; for a function that represents the amount
of material to be used in a manufacturing process, we would be interested in the min—
imum vaiue. In this section we learn how to ﬁnd the maximum and minimum values
of quadratic and other functions. ...
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This note was uploaded on 10/07/2009 for the course MATH 150 taught by Professor Unknown during the Spring '06 term at Ohio State.
 Spring '06
 UNKNOWN
 Math

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