Pre Calculus 40S
Lesson 9.1: Transforming Rational functions
Objectives: Use transformations to Graph and Analyze Rational Functions.
Lesson:
This Chapter is meant to build on the ideas about rational functions introduced in Pre-Calculus 11.
Lets begin recalling a few things:
Example 1:
a. Using a table of values graph .
b. What is the Domain of this graph?
c. What is the Range of this graph?
d. Give an ordered pair that is not on the graph.
ANS
a.
Example 2:
a. Using a table of values, graph y = 1/x2 .
b. What is the Domain of this graph?
c. What is the Range of this graph?
d. Give an ordered pair that is not on the graph.
ANS
a.
Chapter 9 Notes
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Pre Calculus 40S
Example 2: Respond to each of the questions.
1. State at least two common characteristics that all four graphs share.
2. State at least two differences between Graph 3 and Graph 4.
3. Based on the graphs, state the non-permissible values of the related rational equation that describes each
graph.
4. State a specific similarity and a specific difference between Graph 3 and Graph 4.
Example 3: Use the following to help you graph:
Step 1 − Find the Vertical Asymptotes for a graph.
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Pre Calculus 40S
Step 2 − Find the Horizontal Asymptotes for the graph.
Step 3 − Find the intercepts for the graph.
The x-intercept:
The y – intercept:
Step 4 − Find ordered pairs on the graph to determine its behavior.
Chapter 9 Notes
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Pre Calculus 40S
Example 4: Graphing (highest degree of 1)
Draw the following graph accurately:
Step 1 − Find the Vertical Asymptotes for a graph.
Step 2 − Find the Horizontal Asymptotes for the graph.
Step 3 − Find the intercepts for the graph.
The x-intercept:
The y – intercept:
Step 4 − Find ordered pairs on the graph to determine its behavior.
Chapter 9 Notes
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Pre Calculus 40S
Homework 9.1a
Graphing and Analyzing Rational Functions − Check Your Understanding Questions
1. What is the vertical asymptote of the graph of the function
?
2. Find the equation of the horizontal asymptote for the graph of the function
.
3. What is the x-intercept for the graph of the function
?
4. What is the y-intercept for the graph of the function
?
5. What is the x-intercept of the graph of the function
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Pre Calculus 40S
Example 5: Graph the function
.
Step 1 − Find the Vertical Asymptotes for a graph.
Step 2 − Find the Horizontal Asymptotes for the graph.
Step 3 − Find the intercepts for the graph.
The x-intercept:
The y – intercept:
Step 4 − Find ordered pairs on the graph to determine its behavior.
Example 6: Find all the asymptotes on the graph of the function
.
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Pre Calculus 40S
Lets Recall some shapes of quadratic rational graphs:
Shape 1
shape 2
Chapter 9 Notes
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shape 3
Pre Calculus 40S
Example 7: Graph the function
Step 1 − Find the Vertical Asymptotes for a graph.
Step 2 − Find the Horizontal Asymptotes for the graph.
Step 3 − Find the intercepts for the graph.
The x-intercept:
The y – intercept:
Step 4 − Find ordered pairs on the graph to determine its behavior.
Chapter 9 Notes
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Pre Calculus 40S
Example 8: Graph the function
Step 1 − Find the Vertical Asymptotes for a graph.
Step 2 − Find the Horizontal Asymptotes for the graph.
Step 3 − Find the intercepts for the graph.
The x-intercept:
The y – intercept:
Step 4 − Find ordered pairs on the graph to determine its behavior.
Chapter 9 Notes
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Pre Calculus 40S
Example 9: Graph the rational function
Example 10: Graph the rational function
Chapter 9 Notes
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Pre Calculus 40S
Example 11:
a. What is the x-intercept of the graph of
b. What is the solution to the rational equation
c. Describe the relationship between the roots of a rational equation and the x-intercepts of
the graph of the corresponding function.
Example 12: The concentration of a particular drug for treating a disease is given by the equation
where the concentration C is measured in µmg/mL and the time is measured in minutes.
a. Based on this model, at what time is the concentration of the drug the highest in the
bloodstream and what is that concentration?
b. Will the concentration of the drug in the bloodstream ever be 0? Explain.
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Pre Calculus 40S
c. Compare the times for the drug to reach its maximum concentration in the blood and the
time for it to dissipate.
d. What is the significance of the asymptotes for this graph?
e. Although it is not possible to have a negative time, explain why there is an intercept for the
graph along the horizontal asymptote.
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Pre Calculus 40S
Example 13: Determine the equation of the rational function that appears in the graph below.
Chapter 9 Notes
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Pre Calculus 40S
Homework 9.1b
1. Given the function
,
determine the equations of the vertical and horizontal asymptotes (if there are any) and state the value of
the intercepts (if there are any) for this function.
2. Does the graph of the function
have any vertical asymptotes? Explain your answer.
3. Graph the function
and use the graph to approximate the solution to the rational equation
.
Comment on the relationship between the solution to this rational equation and the zero of the function
you graphed.
4. Which equation from the given list describes the equation of the graph below? Explain your answer.
Chapter 9 Notes
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Pre Calculus 40S
5. An astute mathematics teacher determines experimentally that the food consumption of a mouse
population hiding under the dishwasher compared to the food supplied to that population is modeled by
the equation
,
where 'x' is the amount of food supplied and 'y' is the amount of food consumed by the population (both
are measured in grams).
a. Determine the level of consumption that will satisfy the food need of the population.
b. If the teacher is trying to trap all the mice, what amount of food should be supplied to the
population? Explain your reasoning.
6. Graph, by the four step method, the function
.
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Pre Calculus 40S
7. Determine the value that the function
approaches as x → ∞, and as x → -1.
8. Completely analyze the equation, and draw an accurate graph of the function
.
9. Analyze the equation and sketch the graph of the function
.
10. Sketch the graph of
Chapter 9 Notes
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