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Title: Splash Screen

1
Splash Screen
2
Five-Minute Check (over Lesson 2-4) Then/Now New
Vocabulary Key Concept Vertical and Horizontal
Asymptotes Example 1 Find Vertical and
Horizontal Asymptotes Key Concept Graphs of
Rational Functions Example 2 Graph Rational
Functions n lt m and n gt m Example 3 Graph a
Rational Function n m Key Concept Oblique
Asymptotes Example 4 Graph a Rational Function
n m 1 Example 5 Graph a Rational Function
with Common Factors Example 6 Solve a Rational
Equation Example 7 Solve a Rational Equation
with Extraneous Solutions Example 8 Real-World
Example Solve a Rational Equation
3
5Minute Check 1
List all possible rational zeros of f (x) 2x 4
x 3 3x 2 31x 15. Then determine which, if
any, are zeros.
4
5Minute Check 2
List all possible rational zeros of g (x) x 4
x 3 2x 2 4x 8. Then determine which, if
any, are zeros.
5
5Minute Check 3
Describe the possible real zeros of f (x) 2x 5
x 4 8x 3 8x 2 9x 9.
A. 4 positive zeros 1 negative zero B. 4, 2, or
0 positive zeros 1 negative zero C. 3 or 1
positive zeros 1 negative zero D. 2 or 0
positive zeros 2 or 0 negative zeros
6
5Minute Check 4
Describe the possible real zeros of g (x) 3x 4
16x 3 7x 2 64x 20.
A. 1 positive zero 3 or 1 negative zeros B. 1
positive zero 3 negative zeros C. 1 or 0
positive zeros 3 or 1 negative zeros D. 1
positive zero 2 or 0 negative zeros
7
5Minute Check 5
Write a polynomial function of least degree with
real coefficients in standard form that has 2,
5, and 3 i as zeros.
A. f (x) x 4 3x 3 x 2 27x 90 B. f (x)
x 4 3x 3 20x 2 84x 80 C. f (x) x 4 9x
3 18x 2 30x 100 D. f (x) x 4 3x 3 18x
2 90x 100
8
Then/Now
You identified points of discontinuity and end
behavior of graphs of functions using limits.
(Lesson 1-3)
• Analyze and graph rational functions.
• Solve rational equations.

9
Vocabulary
• rational function
• asymptote
• vertical asymptote
• horizontal asymptote
• oblique asymptote or slant asymptote
• holes

10
Key Concept 1
11
Example 1
Find Vertical and Horizontal Asymptotes
Step 1 Find the domain. The function is
undefined at the real zero of the denominator b
(x) x 1. The real zero of b (x) is 1.
Therefore, the domain of f is all real numbers
except x 1.
12
Example 1
Find Vertical and Horizontal Asymptotes
13
Example 1
Find Vertical and Horizontal Asymptotes
Step 1 The zeros of the denominator b (x) 2x2
1 are imaginary, so the domain of f is all real
numbers.
14
Example 1
Find Vertical and Horizontal Asymptotes
15
Example 1
Find Vertical and Horizontal Asymptotes
16
Example 1
17
Key Concept 2
18
Example 2
Graph Rational Functions n lt m and n gt m
Step 2 There is a vertical asymptote at x
5. The degree of the polynomial in the
numerator is 0, and the degree of the polynomial
in the denominator is 1. Because 0 lt 1, the graph
of k has a horizontal asymptote at y 0.
19
Example 2
Graph Rational Functions n lt m and n gt m
Step 3 The function in the numerator has no real
zeros, so k has no x-intercepts. Because k(0)
1.4, the y-intercept is 1.4. Step 4 Graph the
asymptotes and intercepts. Then choose x-values
that fall in the test intervals determined by the
vertical asymptote to find additional points to
plot on the graph. Use smooth curves to complete
the graph.
20
Example 2
Graph Rational Functions n lt m and n gt m
21
Example 2
Graph Rational Functions n lt m and n gt m
22
Example 2
Graph Rational Functions n lt m and n gt m
Step 2 There are vertical asymptotes at x 2 and
x 2. Compare the degrees of the numerator
and denominator. Because 1 lt 2, there is a
horizontal asymptote at y 0. Step 3 The
numerator has a zero at x 1, so the
x-intercept is 1. f(0) ?0.25, so The
y-intercept is 0.25.
23
Example 2
Graph Rational Functions n lt m and n gt m
Step 4 Graph the asymptotes and intercepts. Then
find and plot points in the test intervals
determined by the intercepts and vertical
asymptotes (8, 2), (2, 1), (1, 2), (2, 8).
Use smooth curves to complete the graph.
24
Example 2
Graph Rational Functions n lt m and n gt m
25
Example 2
A. vertical asymptotes x 4 and x 3
horizontal asymptote y 0 y-intercept
0.0833 B. vertical asymptotes x 4 and x 3
horizontal asymptote y 1 intercept
0 C. vertical asymptotes x 4 and x 3
horizontal asymptote y 0 intercept
0 D. vertical asymptotes x 4 and x 3
horizontal asymptote y 1 y-intercept 0.0833
26
Example 3
Graph a Rational Function n m
27
Example 3
Graph a Rational Function n m
Step 3 The x-intercepts are 3 and 4, the zeros
of the numerator. The y-intercept is 1.5 because
f(0) 1.5.
28
Example 3
Graph a Rational Function n m
Step 4 Graph the asymptotes and intercepts. Then
find and plot points in the test intervals (8,
3), (3, 2), (2, 2), (2, 4), (4, 8).
29
Example 3
Graph a Rational Function n m
30
Example 3
A. vertical asymptote x 2 horizontal asymptote
y 6 x-intercept 0.833 y-intercept
2.5 B. vertical asymptote x 2 horizontal
asymptote y 6 x-intercept 2.5 y-intercept
0.833 C. vertical asymptote x 6 horizontal
asymptote y 2 x-intercepts 3 and 0
y-intercept 0 D. vertical asymptote x 6,
horizontal asymptote y 2 x-intercept 2.5
y-intercept 0.833
31
Key Concept 3
32
Example 4
Graph a Rational Function n m 1
Step 2 There is a vertical asymptote at x
3. The degree of the numerator is greater than
the degree of the denominator, so there is no
horizontal asymptote.
33
Example 4
Graph a Rational Function n m 1
Because the degree of the numerator is exactly
one more than the degree of the denominator, f
has an oblique asymptote. Using polynomial long
division, you can write the following.
Therefore, the equation of the oblique/slant
asymptote is y x 2.
34
Example 4
Graph a Rational Function n m 1
Step 4 Graph the asymptotes and intercepts. Then
find and plot points in the test intervals (8,
3.37), (3.37, 3), (3, 2.37), (2.37, 8).
35
Example 4
Graph a Rational Function n m 1
36
Example 4
Graph a Rational Function n m 1
37
Example 4
38
Example 5
Graph a Rational Function with Common Factors
39
Example 5
Graph a Rational Function with Common Factors
Step 2 There is a vertical asymptote at the real
zero of the simplified denominator x 2. There
is a horizontal asymptote at y 1, the ratio of
the leading coefficients of the numerator and
denominator, because the degrees of the
polynomials are equal.
40
Example 5
Graph a Rational Function with Common Factors
41
Example 5
Graph a Rational Function with Common Factors
42
Example 5
A. vertical asymptote at x 2, horizontal
asymptote at y 2 no holes B. vertical
asymptotes at x 5 and x 2 horizontal
asymptote at y 1 hole at (5, 3) C. vertical
asymptotes at x 5 and x 2 horizontal
asymptote at y 1 hole at (5, 0) D. vertical
asymptote at x 2 horizontal asymptote at y
1 hole at (5, 3)
43
• Stop here. Review any slides you did not
understand.

44
Example 6
Solve a Rational Equation
Original Equation
Multiply by the LCD, x 6.
Simplify.
Simplify.
45
Example 6
Solve a Rational Equation
46
Example 6
A. 22 B. 2 C. 2 D. 8
47
Example 7
Solve a Rational Equation with Extraneous
Solutions
The LCD of the expressions is x 1.
x 2 x x 3x 2 Simplify. x 2 3x 2
0 Subtract 3x 2 from each side.
48
Example 7
Solve a Rational Equation with Extraneous
Solutions
(x 2)(x 1) 0 Factor. x 2 or x
1 Solve.
Because the original equation is not defined when
x 1, you can eliminate this extraneous
solution. So, the only solution is 2.
49
Example 7
A. 2, 1 B. 1 C. 2 D. 2, 5
50
Example 8
Solve a Rational Equation
51
Example 8
Solve a Rational Equation
Multiply by the LCD.
6r 24 6r 24 2r 2 32 Simplify. 12r
2r 2 32 Combine like terms. 0 2r 2 12r
32 Subtract 12r from each side. 0 r 2 6r
16 Divide each side by 2.
52
Example 8
Solve a Rational Equation
0 (r 8)(r 2) Factor. r 8 or r
2 Solve.
Because r is the rate of the boat, r cannot be
negative. Therefore, r is 8 miles per hour.