C H A P T E R 2 Polynomial and Rational Functions

260 Chapter 2 Polynomial and Rational Functions
7.
9.
10.
f x x kqx r 8. (a)
(a) Cubic, passes through 2, 5, rises to the right
a bia bi a 2 abi abi b 2 i 2 a 2 b 2
Since a and b are real numbers, a is also a real number.
2 b2 fx
Vertical asymptote:
Horizontal asymptote:
(i)
(ii)
(iii)
(iv)
One possibility:
f x x 2x 2 5
x 3 2x 2 5
(b) Cubic, passes through 3, 1, falls to the right
One possibility:
f x x 3x 2 1
x 3 3x 2 1
ax b
cx d
x d
c
y a
c
a > 0, b < 0, c > 0, d < 0
Both the vertical asymptote and the horizontal
asymptote are positive. Matches graph (d).
a > 0, b > 0, c < 0, d < 0
Both the vertical asymptote and the horizontal
asymptote are negative. Matches graph (b).
a < 0, b > 0, c > 0, d < 0
The vertical asymptote is positive and the horizontal
asymptote is negative. Matches graph (a).
a > 0, b < 0, c > 0, d > 0
The vertical asymptote is negative and the horizontal
asymptote is positive. Matches graph (c).
(b)
(c)
z m 1
z
1 1 1 i
1 i 1 i 1 i
z m 1
z
z m 1
z
1 i
2
1 1 3 i
3 i 3 i 3 i
3 i
10
1
2 8i
1 2 8i
2 8i 2 8i
2 8i
68
1 1
2 2 i
3 1
10 10 i
1
2
34 17 i
11.
(a) is a vertical asymptote.
a causes a vertical stretch if and a vertical
shrink if For a > 1, the graph
becomes wider as a increases. When a is negative
the graph is reflected about the x-axis.
0 < fx
b 0 ⇒ x b
a > 1
a < 1.
ax
x b2 (b) a 0. Varying the value of b varies the vertical
asymptote of the graph of f. For b > 0, the graph
is translated to the right. For b < 0, the graph is
reflected in the x-axis and is translated to the left.