EXERCISE 2-2
Things to remember:
1.
LIBRARY OF ELEMENTARY FUNCTIONS
f(x)
m(x)
h(x)
5
5
5
-5
0
5
-5
x
-5
0
5
x
-5
Square Function
h(x) = x2
Domain: All real numbers
Range: [0, ∞)
(b)
Cube Function
m(x) = x3
Domain: All real numbers
Range: All real numbers
(c)
p(x)
n(x)
5
x
-5
Identity Function
f(x) = x
Domain: All real numbers
Range: All real numbers
(a)
0
g(x)
5
5
5
-5
0
5
0
5
x
x
-5
0
5
x
-5
Square-Root Function
n(x) = x
Domain: [0, ∞)
Range: [0, ∞)
(d)
Cube-Root Function
3
p(x) = x
Domain: All real numbers
Range: All real numbers
(e)
Absolute Value Function
g(x) = |x|
Domain: All real numbers
Range: [0, ∞)
(f)
designate the above functions may vary from
!NOTE: Letters used to !
context to context.
2. GRAPH TRANSFORMATIONS SUMMARY
Vertical Translation:
k > 0 Shift graph of y = f(x) up k units
y = f(x) + k !"
#k < 0 Shift graph of y = f(x) down |k| units
Horizontal Translation:
!h > 0 Shift graph of y = f(x) left h units
y = f(x + h) "
#h < 0 Shift graph of y = f(x) right |h| units
Reflection:
y = -f(x)
Reflect the graph of y = f(x) in the x axis
Vertical Stretch and Shrink:
Stretch graph of y = f(x) vertically by
"
multiplying each ordinate value by A
y = Af(x) $ A > 1
#
$%0 < A < 1 Shrink graph of y = f(x) vertically by
multiplying each ordinate value by A
!
EXERCISE 2-2 39
3.
PIECEWISE-DEFINED FUNCTIONS
Functions whose definitions involve more than one rule are
called PIECEWISE-DEFINED FUNCTIONS.
For example,
$
if x < 0
f(x) = | x | = %"x
& x if x # 0
is a piecewise-defined function.
To graph a piecewise-defined function, graph each rule over
the appropriate portion of the domain.
!
1. f(x) = 2x; domain: all real numbers; range: all real numbers
3. h(x) = -0.6 x ; domain: [0, ∞); range: (-∞, 0]
5. m(x) = 3|x|; domain: all real numbers; range: [0, ∞)
7. r(x) = -x3; domain: all real numbers; range: all real numbers
!
9.
11.
13.
15.
17.
19.
21. The graph of g(x) = -|x + 3| is
the graph of y = |x| reflected
in the x axis and shifted 3
units to the left.
40
CHAPTER 2
FUNCTIONS AND GRAPHS
23. The graph of f(x) =
(x - 4)2 - 3 is the graph of
y = x2 shifted 4 units to the
right and 3 units down.
25. The graph of f(x) = 7 - x is
the graph of y = x reflected
in the x axis and shifted 7
units up.
!
!
27. The graph of h(x) = -3|x| is
the graph of y = |x| reflected
in the x axis and vertically
expanded by a factor of 3.
29. The graph of the basic function y = x2 is shifted 2 units to the left
and 3 units down. Equation: y = (x + 2)2 - 3.
31. The graph of the basic function y = x2 is reflected in the x axis,
shifted 3 units to the right and 2 units up.
Equation: y = 2 - (x - 3)2.
33. The graph of the basic function y = x is reflected in the x axis and
shifted 4 units up. Equation: y = 4 - x .
35. The graph of the basic function y = x3 is shifted 2 units to the left
and 1 unit down. Equation: y =! (x + 2)3 - 1.
!
37. g(x) = x " 2 - 3
39. g(x) = -|x + 3|
41. g(x) = -(x - 2)3 - 1
!
f
5
43.
x
x
45.
30
47.
100
80
20
–5
5
60
x
40
10
20
–5
y
10
20
20
40
60
80
y
49. The graph of the basic function: y = |x| is reflected in the x axis
and has a vertical contraction by the factor 0.5.
Equation: y = -0.5|x|.
EXERCISE 2-2 41
51. The graph of the basic function y = x2 is reflected in the x axis and
is vertically expanded by the factor 2. Equation: y = -2x2.
53. The graph of the basic function y = 3 x is reflected in the x axis and
is vertically expanded by the factor 3. Equation: y = -3 3 x .
55. Vertical shift, horizontal shift.
!
Reversing the order does not change
the result. Consider a point
!
(a, b) in the plane. A vertical shift of k units followed
by a
horizontal shift of h units moves (a, b) to (a, b + k) and then to
(a + h, b + k).
In the reverse order, a horizontal shift of h units followed by a
vertical shift of k units moves (a, b) to (a + h, b) and then to
(a + h, b + k). The results are the same.
57. Vertical shift, reflection in the x axis.
Reversing the order can change the result. For example, let (a, b) be
a point in the plane with b > 0. A vertical shift of k units, k ≠ 0,
followed by a reflection in the x axis moves (a, b) to (a, b + k) and
then to (a, -[b + k]) = (a, -b - k).
In the reverse order, a reflection in the x axis followed by the
vertical shift of k units moves (a, b) to (a, -b) and then to
(a, -b + k); (a, -b - k) ≠ (a, -b + k) when k ≠ 0.
59. Horizontal shift, reflection in y axis.
Reversing the order can change the result. For example, let (a, b) be
a point in the plane with a > 0. A horizontal shift of h units
followed by a reflection in the y axis moves (a, b) to the point
(a + h, b) and then to (-[a + h], b) = (-a - h, b).
In the reverse order, a reflection in the y axis followed by the
horizontal sift of h units moves (a, b) to (-a, b) and then the
(-a + h, b), (-a - h, b) ≠ (-a + h, b) when h ≠ 0.
61. (A) The graph of the basic
function y = x is reflected
in the x axis, vertically
expanded by a factor of 4, and
shifted up 115 units.
!
(B)
42
CHAPTER 2
FUNCTIONS AND GRAPHS
63. (A) The graph of the basic
function y = x3 is vertically
contracted by a factor of
0.00048 and shifted right 500
units and up 60,000 units.
(B)
8.50 + 0.0650x
if
65. (A) S(x) = "#
$8.50 + 0.0650(700) + 0.09(x ! 700) if
8.50 + 0.0650x if 0 ! x ! 700
= "#
if x > 700
$!9 + 0.09x
0 ! x ! 700
x > 700
S
(B)
100
50
700
x
67. (A) If 0 ≤ x ≤ 30,000, T(x) = 0.035x, and T(30,000) = 1,050.
If 30,000 < x ≤ 60,000, T(x) = 1,050 + 0.0625(x - 30,000)
= 0.0625x - 825, and
T(60,000) = 2,925.
If x > 60,000, T(x) = 2,925 + 0.0645(x - 60,000)
= 0.0645x - 945.
$&0.035x
if 0 " x " 30, 000
Thus, T(x) = %0.0625x # 825 if 30, 000 < x " 60, 000
&'0.0645x # 945 if
x > 60, 000
T
(B)
(C) T(40,000) = 0.0625(40,000) - 825
= 1,675;
$1,675
4,000
3,000
!
2,000
1,000
60,000
x
69. (A) The graph of the basic
function y = x is vertically
expanded by a factor of 5.5 and
shifted down 220 units.
T(70,000) = 0.0645(70,000) - 945
= 3,570;
$3,570
71. (A) The graph of the basic
function y = x is vertically
expanded by a factor of 7.08.
function y = x is vertically expanded by
!
(B)
(B)
EXERCISE 2-2 43