Algebra II
Unit #5: Rational Functions
(8.1-8.2)
Name: ________________________
Block: __________
Direct Variation
Functions that have a direct variation take the form y = kx, k ≠ 0 where k is the
constant of variation. What will the graph of all direct variation relationships look like?
Give an example to support your answer.
In a direct variation relationship, as x increases, y __________________ proportionally. (assuming positive)
In a direct variation relationship, as x decreases, y __________________ proportionally. (assuming positive)
**The ratio of y to x should be CONSTANT!!
Determine whether the data shows a direct variation. If it does, find the k value and write a function to model
the data. Show your work.
a. (1,2) (2,4) (6, 18) (25, 50)
b. (24,6) (32, 8) (40, 10) (8,2)
c.
X
Y
3
27
5
45
6
54
9
81
Inverse Variation
What do you think of when you hear the word inverse? ____________________________________________
In an inverse variation relationship, as x increases, y __________________ proportionally.
In an inverse variation relationship, as x decreases, y __________________ proportionally.
Inverse variation equations can take multiple forms: (k is the constant of variation)
**Most Common (we will use): ____________
Others: ________ and _________
To test if data shows an inverse relationship, use ___________________ (this also finds k!)
Determine whether the data shows an inverse variation. If it does, find the k value and write a function in the
𝑘
form 𝑦 = 𝑥 to model the data. Show your work.
a. (2,20) (5,8) (4,10)
b. (3,10) (4,8) (10,3) (15, 2)
c.
X
Y
1
50
2
25
5
10
10
5
1
Real-Life Application: Given the following relationships below, determine if the scenario has the potential to
be a direct variation relationship (D) or an inverse variation relationship (I).
x
y
Potential
relationship
a)
Temperature
Time it takes a block of ice to melt
b)
Number of hours you study
Grade (%) on a test
c)
How many hours you work
How much money you make
d)
Average speed of a car
Time it takes to get to school
e)
Price of a ticket
Amount of tickets that you can purchase
f)
Number of people living in an apartment
Cost per person to rent the apartment
g)
Initial height of a ball
Time it takes for the ball to hit the ground
h)
Depth of the ocean
Temperature of the ocean
i)
Speed of a Car
Rate of Gas Consumption
j)
Radius of a pizza
The Area of a Pizza
Graphing Inverse Variations: Evaluate the inverse functions for the given values to complete each table.
Then plot your points on the axes.
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a. 𝑦 = 𝑥
X
Y
b. 𝑥 =
X
1
Y
X
-1
X
10
-1
2
-2
5
-4
5
4
-1
2
0
10
c. 𝑥𝑦 = −100
𝑦
Y
2
1
10
Y
X
-2
0
Y
Y
-2
20
-10
25
X
-10
0
What does the graph of an inverse variation relationship look like? Describe the shape, give specific features!
2
Writing: Inverse Restrictions: Why can x and y never be zero (𝑥 ≠ 0, 𝑦 ≠ 0) in an inverse relationship? How
can you see this on the graphs of an inverse relationship? **We will return to graphing in 8.2**
Determine whether the relationship between the variables is a direct variation, an inverse variation, or neither.
Write a function model for direct or inverse variations.
Writing: How can you tell whether a set of data varies directly or inversely? (consider a-c as examples)
Modeling Inverse Relationships
#1) Our Algebra 2 class has decided to pick up litter each weekend at Bethel Park. Each week there is approximately the
same amount of litter at Bethel Park. The table below shows the number of students who worked each week and the time
needed for the pickup.
Number of students (n)
3
5
10
17
Time in minutes (t)
85
51
25.5
15
a) Plot the data on the graph provided. Label the axes
with appropriate variables.
b) What things do you look for to identify if it COULD
be an inverse relationship?
c) Determine the constant of variation by examining
each ordered pair. Show you work!
d) Write a function to model this data (use the given variables): _____________________.
e) Use your function from (d) to determine the number of students needed to complete the project in at most 30
minutes each week. Show your work below. Identify this solution on the graph.
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#2) Suppose you have $200 total from your summer savings. You want to treat you and your friends to the
Garth Brooks concert. Let p be the price of the ticket and n be the number of tickets you can buy.
a) Fill in the table below and plot the points on the graph
provided. Label your axes with the appropriate
variables.
IV:
5
10
20
25
50
100
DV:
b) Write a function that would determine n, the number of
tickets you can buy for any price p. Use function
notation.
c) Use your function to determine the number of tickets that you can buy if the price of the tickets is $24
each. Show your work.
#3) Suppose that x and y vary inversely and that 𝑥 = 15 when 𝑦 = 3.
(a) Write a function rule that models the inverse variation. ____________
(b) Use the function from part (a) to find the value of y when 𝑥 = 9. Show your work.
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(c) Use the function from part (a) to find the value of x when 𝑦 = 4. Show your work.
#4) Suppose y varies inversely with x such that 𝑦 = 8 when 𝑥 = 7.
(a) Write a function rule that models the inverse variation. ____________
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(b) Use the function from part (a) to find the value of y when 𝑥 = 2. Show your work.
(c) Use the function from part (a) to find the value of x when 𝑦 = 4. Show your work.
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8.2 The Reciprocal Function Family
Vocabulary Review: “Reciprocal”
The Reciprocal Function Family
Functions that model inverse variation belong to the reciprocal function family. They take the form
𝑎
𝑓(𝑥) = 𝑥−ℎ + 𝑘. In this form, 𝑥 ≠ ℎ because _______________________________. The parent function is
𝟏
𝒇(𝒙) = 𝒙 where 𝒙 ≠ 𝟎. Graph the reciprocal parent function by completing the tables and plotting the points.
x
-20
-16
-12
-8
-4
-2
-1
1
2
4
8
12
16
20
y
x
y
a. Examine the table to see what the y value is when x = 0. Why does this happen?
b. Examine both negative and positive values of x in your calculator. Describe what happens to the yvalues as x approaches zero. (To really see this, set up your table to count by 0.01 or 0.001).
c. What happens to the y-values as x increases?
d.
What happens to the y-values as x decreases?
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KEY FEATURES / VOCABULARY
**Each part of the graph of a reciprocal function is called a _______________.
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Which quadrants are the branches of the parent function 𝑓(𝑥) = 𝑥 in? ____ and ____
x-intercepts: _____________________
y-intercept: _____________________________
What is an Asymptote? A line such that the distance between the function (actual curve) and the line
approaches 0 as they tend to infinity.
In my words:
Vertical Asymptote: ___________________
Horizontal Asymptote: _______________________
Domain (all possible x values for which the function is defined): _____________________________
Range: (all possible y values for which the function is defined): ______________________________
Graph Exploration – Transformations of Reciprocal Functions
** use different colors for each function on the graph and label them**
1
Part I: Using your calculator, graph each function on the coordinate plane (𝑦 = 𝑥 is already graphed).
Label these on the graph and use a different color for each. Compare the graphs to determine the effect of
changing the a value.
5
x
10
y
x
0 .5
y
x
1
y
x
 10
y
x
a) y 
b)
c)
d)
e)
Transformation: How would you summarize the
effect of the value of a on the graph? Be specific.
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Part II: Using your calculator, graph each function on the coordinate plane with the parent function.
Identify both asymptotes and describe the transformation caused by changing the h value.
1
a) 𝑓(𝑥) = 𝑥−2 , 𝑥 ≠ ____
horizontal asymptote: ____________
vertical asymptote: ____________
1
b) 𝑓(𝑥) = 𝑥+4 , 𝑥 ≠ _____
horizontal asymptote: ____________
vertical asymptote: ____________
Transformation: How would you summarize the
effect of the value of h on the graph? Be specific.
Part III: Using your calculator, graph each function on the coordinate plane with the parent function.
Identify both asymptotes and describe the transformation caused by changing the k value.
1
𝑥
a) 𝑓(𝑥) = + 5 , 𝑥 ≠ 0
horizontal asymptote: ____________
vertical asymptote: ____________
1
b) 𝑓(𝑥) = 𝑥 − 3 , 𝑥 ≠ 0
horizontal asymptote: ____________
vertical asymptote: ____________
Transformation: How would you summarize the
effect of the value of k on the graph? Be specific.
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Writing: Summarize ALL transformation effect of the a, h, and k values on a reciprocal function in the
𝑎
form 𝑓(𝑥) = 𝑥−ℎ + 𝑘 where 𝑥 ≠ ℎ
a
h
k
Writing: How is transforming a reciprocal function similar to and different from transforming a
quadratic function? Give specific comparisons.
Writing: Domain and Range What restrictions will reciprocal functions have in the domain and range?
What causes these “restrictions?” How can you identify these restrictions on the graph and when writing
the domain or range?
Multiple Transformations
1
 6 ? Identify the domain and range.
x4
a) Find the equations of the asymptotes. The sketch the asymptotes on the graph.
1. What is the graph of y 
The Vertical asymptote is x = _______.
The Horizontal asymptote is y = _____.
b) Translate the points (1,1) and (-1, -1) from the
1
graph of y  . Each point moves 4 units to
x
the _____ and 6 units ______.
c) Draw the branches of y 
1
 6 through
x4
translated points.
d) Find the domain and range.
Domain: the set of all real numbers except
x = ______.
Range: the set of all real numbers except
y = _______.
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3
 2 ? Identify the domain and range. (the parent graph is shown)
x 1
a) Find the equations of the asymptotes. The sketch the asymptotes on the graph.
2. What is the graph of y 
The vertical asymptote is x = _______.
The horizontal asymptote is y = ___________.
b) FIRST, Sketch the graph of y 
3
.
x
c) Then, translate the points from the graph of y 
3
.
x
Each point moves 1 unit to the _______ and
2 units _______.
d) Draw the branches of y 
3
 2 through
x 1
translated points.
e) Find the domain and range.
Domain: the set of all real numbers except x = ______.
Range: the set of all real numbers except y = _______.
3. Sketch the asymptotes and the graph of each function. Identify the domain and range.
a) y 
1
3
x
D: __________________________
R: __________________________
b) y 
1
7
x
D: ____________________________
R: ____________________________
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c)
f ( x) 
1
5
x2
f ( x) 
d)
D: __________________________
R: __________________________
2
1
x6
D: ____________________________
R: ____________________________
4. This graph of a function is a translation of the graph of y 
2
. What is an equation for the function?
x
a) Identify the asymptotes: x = ______ and y = ______.
So h = ______ and k = ______.
b) Substitute the a, h, and k values into the general form to write the
equation to model the graph.
5. Write an equation for the translation of y 
1
that has the given asymptotes and other features.
x
a) Stretched by 2, VA at 𝑥 = 0, HA at 𝑦 = 4 __________________________
b) Reflected across the x axis, VA at 𝑥 = −3, HA at 𝑦 = 7 __________________________
c) Reflected across the x-axis, stretched by 6, VA at 𝑥 = 4, HA at 𝑦 = −8 ________________________
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