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Holt Algebra 1
12-2 Rational Functions12-2 Rational Functions
Holt Algebra 1
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
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Holt Algebra 1
12-2 Rational Functions
Warm Up
The inverse variation xy = 8 relates the constant speed x in mi/h to the time y in hours that it takes to travel 8 miles. Graph this inverse variation. Then use the graph to estimate how many hours it would take to travel 8 miles jogging at a speed of 4.5 mi/h.
Possible answer: 1 h 45 min
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Holt Algebra 1
12-2 Rational Functions
Identify excluded values of rational functions.
Graph rational functions.
Objectives
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Holt Algebra 1
12-2 Rational Functions
rational functionexcluded valuediscontinuous functionasymptote
Vocabulary
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Holt Algebra 1
12-2 Rational Functions
A rational function is a function whose rule is a quotient of polynomials in which the denominator has a degree of at least 1. In other words, there must be a variable in the denominator. The inverse variations you studied in the previous lesson are a special type of rational function.
Rational functions:
Not rational functions:
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Holt Algebra 1
12-2 Rational Functions
For any function involving x and y, an excluded value is any x-value that makes the function value y undefined. For a rational function, an excluded value is any value that makes the denominator equal to 0.
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Holt Algebra 1
12-2 Rational Functions
Example 1: Identifying Excluded Values
Identify the excluded value for each rational function.
A.
x = 0 Set the denominator equal to 0.The excluded value is 0.
B.
x – 2 = 0
x = 2
Set the denominator equal to 0.Solve for x.
The excluded value is 2.
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Holt Algebra 1
12-2 Rational Functions
Check It Out! Example 1
Identify the excluded value for each rational function.
a.
x = 0 Set the denominator equal to 0.The excluded value is 0.
b.
x – 1 = 0
x = 1
Set the denominator equal to 0.Solve for x.
The excluded value is 1.
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Holt Algebra 1
12-2 Rational Functions
Check It Out! Example 1
Identify the excluded value for each rational function.
x + 4 = 0
x = –4
Set the denominator equal to 0.Solve for x.
The excluded value is –4.
c.
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Holt Algebra 1
12-2 Rational Functions
Most rational functions are discontinuous functions, meaning their graphs contain one or more jumps, breaks, or holes. This occurs at an excluded value.
One place that a graph of a rational function is discontinuous is at an asymptote. An asymptote is a line that a graph gets closer to as the absolute value of a variable increases. In the graph shown, both the x- and y-axes are asymptotes. The graphs of rational functions will get closer and closer to but nevertouch the asymptotes.
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Holt Algebra 1
12-2 Rational Functions
For rational functions, vertical asymptotes will occur at excluded values.
Look at the graph of y = The denominator is 0 when x = 0 so 0 is an excluded value. This means there is a vertical asymptote at x = 0. Notice the horizontal asymptote at y = 0.
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Holt Algebra 1
12-2 Rational Functions
Vertical lines are written in the form x = b, and horizontal lines are written in the form y = c.
Writing Math
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Holt Algebra 1
12-2 Rational Functions
Look at the graph of y = Notice that the graph of the parent function y= has been translated 3 units right and there is a vertical asymptote at x = 3. The graph has also been translated 2 units up and there is a horizontal asymptote at y = 2.
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Holt Algebra 1
12-2 Rational Functions
These translations lead to the following formulas for identifying asymptotes in rational functions.
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Holt Algebra 1
12-2 Rational Functions
Example 2A: Identifying Asymptotes
Identify the asymptotes.
Step 2 Identify the asymptotes.
vertical: x = –7
horizontal: y = 0
Step 1 Write in y = form.
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Holt Algebra 1
12-2 Rational Functions
Example 2B: Identifying Asymptotes
Identify the asymptotes.
Step 1 Identify the vertical asymptote.
2x – 3 = 0
+3 +32x = 3
Find the excluded value. Set the denominator equal to 0.
Add 3 to both sides.
Solve for x. Is an excluded value.
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Holt Algebra 1
12-2 Rational Functions
Example 2B Continued
Identify the asymptotes.
Step 2 Identify the horizontal asymptote.
c = 8y = 8
Vertical asymptote: x = ; horizontal asymptote: y = 8
y = c
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Holt Algebra 1
12-2 Rational Functions
Check It Out! Example 2a
Identify the asymptotes.
Step 1 Identify the vertical asymptote.
x – 5 = 0
+5 +5x = 5
Find the excluded value. Set the denominator equal to 0.
Add 5 to both sides.
Solve for x. 5 is an excluded value.x = 5
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Holt Algebra 1
12-2 Rational Functions
Identify the asymptotes.
Step 2 Identify the horizontal asymptote.
c = 0y = 0 y = c
Check It Out! Example 2a Continued
Vertical asymptote: x = 5; horizontal asymptote: y = 0
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Holt Algebra 1
12-2 Rational Functions
Check It Out! Example 2b
Identify the asymptotes.
Step 1 Identify the vertical asymptote.
4x + 16 = 0
–16 –164x = –16
Find the excluded value. Set the denominator equal to 0.
Subtract 16 from both sides.
Solve for x. –4 is an excluded value.x = –4
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Holt Algebra 1
12-2 Rational Functions
Check It Out! Example 2b Continued
Identify the asymptotes.
Step 2 Identify the horizontal asymptote.
c = 5y = 5
Vertical asymptote: x = –4; horizontal asymptote: y = 5
y = c
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Holt Algebra 1
12-2 Rational Functions
Check It Out! Example 2c
Identify the asymptotes.
Step 1 Identify the vertical asymptote.
x + 77 = 0
–77 –77x = –77
Find the excluded value. Set the denominator equal to 0.
Subtract 77 from both sides.
Solve for x. –77 is an excluded value.
x = –77
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Holt Algebra 1
12-2 Rational Functions
Check It Out! Example 2c Continued
Identify the asymptotes.
Step 2 Identify the horizontal asymptote.
c = –15y = –15
Vertical asymptote: x = –77; horizontal asymptote: y = –15
y = c
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Holt Algebra 1
12-2 Rational Functions
To graph a rational function in the form y =when a = 1, you can graph the asymptotes and then translate the parent function y = . However, if a ≠ 1, the graph is not a translation of the parent function. In this case, you can use the asymptotes and a table of values.
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Holt Algebra 1
12-2 Rational Functions
Example 3A: Graphing Rational Functions Using Asymptotes
Graph the function.
Since the numerator is not 1, use the asymptotes and a table of values.
Step 1 Identify the vertical and horizontal asymptotes.
vertical: x = 3
horizontal: y = 0
Use x = b. x – 3 = 0, so b = 3.
Use y = c. c = 0
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Holt Algebra 1
12-2 Rational Functions
Example 3A Continued
Step 2 Graph the asymptotes using dashed lines.
Step 3 Make a table of values. Choose x-values on both sides of the vertical asymptote.
y
654210x
x = 3
y = 0
Step 4 Plot the points and connect them with smooth curves. The curves will get very close to the asymptotes, but will not touch them.
●●●
●●●
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Holt Algebra 1
12-2 Rational Functions
Example 3B: Graphing Rational Functions Using Asymptotes
Graph the function.
Step 1 Since the numerator is 1, use the asymptotes and translate y = .
vertical: x = –4
horizontal: y = –2
Use x = b. b = –4
Use y = c. c = –2
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Holt Algebra 1
12-2 Rational Functions
Step 2 Graph the asymptotes using dashed lines.
Example 3B Continued
Step 3 Draw smooth curves to show the translation.
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Holt Algebra 1
12-2 Rational Functions
Check It Out! Example 3a
Graph each function.
Step 1 Since the numerator is 1, use the asymptotes and translate y = .
vertical: x = –7
horizontal: y = 3
Use x = b. b = –7
Use y = c. c = 3
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Holt Algebra 1
12-2 Rational Functions
Check It Out! Example 3a Continued
Graph each function.
Step 2 Graph the asymptotes using dashed lines.
Step 3 Draw smooth curves to show the translation.
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Holt Algebra 1
12-2 Rational Functions
Check It Out! Example 3b
Graph each function.
Since the numerator is not 1, use the asymptotes and a table of values.
Step 1 Identify the vertical and horizontal asymptotes.
vertical: x = 3
horizontal: y = 2
Use x = b. x – 3 = 0, so b = 3.
Use y = c. c = 2
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Holt Algebra 1
12-2 Rational Functions
Check It Out! Example 3b
Step 2 Graph the asymptotes using dashed lines.
Step 3 Make a table of values. Choose x-values on both sides of the vertical asymptote.
y
654210x
4
Step 4 Plot the points and connect them with smooth curves. The curves will get very close to the asymptotes, but will not touch them.
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Holt Algebra 1
12-2 Rational Functions
Example 4: Application
Your club has $75 with which to purchase snacks to sell at an afterschool game. The number of snacks y that you can buy, if the average price of the snacks is x-dollars, is given by y =
a. Describe the reasonable domain and range values.
Both the number of snacks purchased and their cost will be positive values so nonnegative values are reasonable for both domain and range.
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Holt Algebra 1
12-2 Rational Functions
Example 4 Continued
b. Graph the function.
Step 1 Identify the vertical and horizontal asymptotes.
vertical: x = 0
horizontal: y = 0
Use x = b. b = 0
Use y = c. c = 0
Step 2 Graph the asymptotes using dashed lines. The asymptotes will be the x- and y-axes.
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Holt Algebra 1
12-2 Rational Functions
Example 4 Continued
Step 3 Since the domain is restricted to nonnegative values, only choose x-values on the right side of the vertical asymptote.
Number of snacks 2 4 6 8Cost of snacks($) 37.5 18.75 12.5 9.38
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Holt Algebra 1
12-2 Rational Functions
Example 4 Continued
Step 4 Plot the points and connect them with a smooth curve.
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Holt Algebra 1
12-2 Rational Functions
Check It Out! Example 4
a. Describe the reasonable domain and range values.
The domain would be all values greater than 0 up to $500 dollars and the range would be all natural numbers greater than 10.
A librarian has a budget of $500 to buy copies of a software program. She will receive 10 free copies when she sets up an account with the supplier. The number of copies y of the program that she can buy is given by
y = + 10, where x is the price per copy.
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Holt Algebra 1
12-2 Rational Functions
Check It Out! Example 4 Continued
b. Graph the function.
Step 1 Identify the vertical and horizontal asymptotes.
vertical: x = 0
horizontal: y = 10
Use x = b. b = 0
Use y = c. c = 10
Step 2 Graph the asymptotes using dashed lines. The asymptotes will be the x- and y-axes.
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Holt Algebra 1
12-2 Rational Functions
Step 3 Since the domain is restricted to nonnegative values, only choose x-values on the right side of the vertical asymptote.
Check It Out! Example 4 Continued
Number of copies 20 40 60 80Price ($) 35 23 18 16
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Holt Algebra 1
12-2 Rational Functions
Step 4 Plot the points and connect them with a smooth curve.
Check It Out! Example 4 Continued
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Holt Algebra 1
12-2 Rational Functions
The table shows some of the properties of the four families of functions you have studied and their graphs.
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Holt Algebra 1
12-2 Rational Functions
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Holt Algebra 1
12-2 Rational Functions
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Holt Algebra 1
12-2 Rational Functions
Lesson Quiz: Part I
Identify the exceeded value for each rational function.
1. 2.
3. Identify the asymptotes of and then graph the function.
x = –4; y = 0
0 5
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Holt Algebra 1
12-2 Rational Functions
Lesson Quiz: Part II
You have $ 100 to spend on CDs. A CD club advertises 6 free CDs for anyone who becomes a member. The number of CDs y
that you can receive is given by y = , where x is the average price per CD.
a. Describe the reasonable domain and range values.D: x > 0
R: natural numbers > 6
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Holt Algebra 1
12-2 Rational Functions
Lesson Quiz: Part III
b. Graph the function.