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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

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.

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.

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

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

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

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.

Vocabulary

- rational function
- asymptote
- vertical asymptote
- horizontal asymptote
- oblique asymptote or slant asymptote
- holes

Key Concept 1

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.

Example 1

Find Vertical and Horizontal Asymptotes

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.

Example 1

Find Vertical and Horizontal Asymptotes

Example 1

Find Vertical and Horizontal Asymptotes

Example 1

Key Concept 2

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.

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.

Example 2

Graph Rational Functions n lt m and n gt m

Example 2

Graph Rational Functions n lt m and n gt m

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.

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.

Example 2

Graph Rational Functions n lt m and n gt m

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

Example 3

Graph a Rational Function n m

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.

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).

Example 3

Graph a Rational Function n m

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

Key Concept 3

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.

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.

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).

Example 4

Graph a Rational Function n m 1

Example 4

Graph a Rational Function n m 1

Example 4

Example 5

Graph a Rational Function with Common Factors

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.

Example 5

Graph a Rational Function with Common Factors

Example 5

Graph a Rational Function with Common Factors

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)

- Stop here. Review any slides you did not

understand.

Example 6

Solve a Rational Equation

Original Equation

Multiply by the LCD, x 6.

Simplify.

Quadratic Formula

Simplify.

Example 6

Solve a Rational Equation

Example 6

A. 22 B. 2 C. 2 D. 8

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.

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.

Answer 2

Example 7

A. 2, 1 B. 1 C. 2 D. 2, 5

Example 8

Solve a Rational Equation

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.

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.

Answer 8

Example 8

A. 1.7 or 8.3 seconds B. 2 or 7 seconds C. 4.7

seconds D. 12 seconds

End of the Lesson