holt mcdougal algebra 2 5-4 rational functions lesson plan mae4945 lesson plan #3 graphing rational...

Holt McDougal Algebra 2

5-4 Rational Functions

Lesson Plan

MAE4945 Lesson Plan #3

Graphing Rational Functions (Cont.)

Joseph Torres

April 9, 2015

Cooperating Teacher Chris Bayus

Observer Professor Susan Steege



Quick Review



The rational function f(x) = can be transformed

by using methods similar to those used to

transform other types of functions.

1x



Like logarithmic and

exponential functions,

rational functions may

have asymptotes. The

function f(x) = has a

vertical asymptote at

x = 0 and a horizontal

asymptote at y = 0.

1x



Factor the numerator.

Identify the zeros and asymptotes of the function. Then graph.

4x – 12 x – 1

f(x) =

4(x – 3) x – 1

f(x) =

The numerator is 0 when x = 3.The denominator is 0 when x = 1.

Zero: 3

Vertical asymptote: x = 1

Horizontal asymptote: y = 4

Example 4C: Graphing Rational Functions with Vertical and Horizontal Asymptotes

The horizontal asymptote is

y =

= = 4. 4 1

leading coefficient of p leading coefficient of q



Factor the numerator and the denominator.

Identify the zeros and asymptotes of the function. Then graph.

3x2 + x x2

– 9 f(x) =

x(3x – 1) (x – 3) (x + 3)

f(x) =

The denominator is 0 when x = ±3.

Vertical asymptote: x = –3, x = 3

Horizontal asymptote: y = 3The horizontal asymptote is

y =

= = 3. 3 1

leading coefficient of p leading coefficient of q

Check It Out! Example 4c

The numerator is 0 when x = 0 or x = – .1

3

Zeros: 0 and – 1 3



In some cases, both the numerator and the denominator of a rational function will equal 0 for a particular value of x. As a result, the function will be undefined at this x-value. If this is the case, the graph of the function may have a hole. A hole is an omitted point in a graph.



Example 5: Graphing Rational Functions with Holes

(x – 3)(x + 3)x – 3

f(x) =

Identify holes in the graph of f(x) = . Then graph.

x2 – 9 x – 3


The expression x – 3 is a factor of both the numerator and the denominator.

There is a hole in the graph at x = 3.

Divide out common factors.

(x – 3)(x + 3)(x – 3)

For x ≠ 3,

f(x) = = x + 3



Example 5 Continued

The graph of f is the same as the graph of y = x + 3, except for the hole at x = 3. On the graph, indicate the hole with an open circle. The domain of f is {x|x ≠ 3}.

Hole at x = 3



Check It Out! Example 5

(x – 2)(x + 3)x – 2

f(x) =

Identify holes in the graph of f(x) = . Then graph.

x2 + x – 6 x – 2


The expression x – 2 is a factor of both the numerator and the denominator.

There is a hole in the graph at x = 2.

Divide out common factors.

For x ≠ 2,

f(x) = = x + 3(x – 2)(x + 3)(x – 2)



Check It Out! Example 5 Continued

The graph of f is the same as the graph of y = x + 3, except for the hole at x = 2. On the graph, indicate the hole with an open circle. The domain of f is {x|x ≠ 2}.

Hole at x = 2



• Real World Example



• Assignment:

• -Vertical Asymptote

• -Horizontal Asymptote

• -Holes

• -Graph

• -Domain

holt mcdougal algebra 2 5-4 rational functions lesson plan mae4945 lesson plan #3 graphing rational...

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