Precalculus Lesson 2.6 – Lesson 2.7 Rational Functions, Asymptotes & Graphs of Rational Functions.
Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational...
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Transcript of Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational...
![Page 1: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/1.jpg)
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
![Page 2: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/2.jpg)
Holt McDougal Algebra 2
5-4 Rational Functions
![Page 3: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/3.jpg)
Holt McDougal Algebra 2
5-4 Rational Functions
Quick Review
![Page 4: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/4.jpg)
Holt McDougal Algebra 2
5-4 Rational Functions
The rational function f(x) = can be transformed
by using methods similar to those used to
transform other types of functions.
1x
![Page 5: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/5.jpg)
Holt McDougal Algebra 2
5-4 Rational Functions
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
![Page 6: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/6.jpg)
Holt McDougal Algebra 2
5-4 Rational Functions
![Page 7: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/7.jpg)
Holt McDougal Algebra 2
5-4 Rational Functions
![Page 8: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/8.jpg)
Holt McDougal Algebra 2
5-4 Rational Functions
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
![Page 9: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/9.jpg)
Holt McDougal Algebra 2
5-4 Rational Functions
![Page 10: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/10.jpg)
Holt McDougal Algebra 2
5-4 Rational Functions
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
![Page 11: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/11.jpg)
Holt McDougal Algebra 2
5-4 Rational Functions
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.
![Page 12: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/12.jpg)
Holt McDougal Algebra 2
5-4 Rational Functions
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
Factor the numerator.
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
![Page 13: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/13.jpg)
Holt McDougal Algebra 2
5-4 Rational Functions
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
![Page 14: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/14.jpg)
Holt McDougal Algebra 2
5-4 Rational Functions
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
Factor the numerator.
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)
![Page 15: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/15.jpg)
Holt McDougal Algebra 2
5-4 Rational Functions
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
![Page 16: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/16.jpg)
Holt McDougal Algebra 2
5-4 Rational Functions
• Real World Example
![Page 17: Holt McDougal Algebra 2 5-4 Rational Functions Lesson Plan MAE4945 Lesson Plan #3 Graphing Rational Functions (Cont.) Joseph Torres April 9, 2015 Cooperating.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649efe5503460f94c12285/html5/thumbnails/17.jpg)
Holt McDougal Algebra 2
5-4 Rational Functions
• Assignment:
• -Vertical Asymptote
• -Horizontal Asymptote
• -Holes
• -Graph
• -Domain