5.8a (Day 1) Absolute Value Function Graphs
Algebra 1
Name ____________________________________
Previously this year we have studied linear equations, y = mx + b. Linear equations, while interesting,
only produce straight lines that either increase at a constant rate or decrease at a constant rate
(or in the special case of y = a constant or x = a constant, don’t increase at all!). Even though linear
functions, are useful, for many students they are no longer exciting. So let’s learn about a second
type of functions….absolute value functions.
Activity 1: Graphing the Parent Function
1. Let’s start with the simplest absolute value function: y = |x|. This function, f(x) = |x|, is called
the “parent function” of the absolute value functions as all other absolute value functions come
from it, or build from it.
a.
Recall that absolute values are distance from 0, and therefore always positive.
Fill in the following table for f(x) = |x|.
b. Graph the resulting ordered pairs.
X
-4
-3
-2
-1
0
1
2
3
4
c.
f(x)
Using a ruler, draw in the graph suggested by your points. Also, add arrows to the ends because
we can assume this pattern will continue indefinitely.
d. What letter of the alphabet would you say the absolute value graph resembles?_______
Why does it have this pattern?
e.
The vertex of the graph is the point at which the y-values change direction; they were
decreasing, and then they begin to increase (or vice-versa). What is the coordinate of the vertex
of the graph f(x) = |x|?
f.
The domain for f(x) = |x| is “all real numbers” because any number you can think of can be put
into this equation for x. You could write the domain as a compound inequality: -∞ < x < +∞.
“All real numbers” is not the range for f(x) = |x| because I can think of many y values that won’t
work in this function (like -1, -12 etc). What is the range of this function?
h. For what value(s) of x will f(x) = 8? Why does your answer make sense in terms of the
definition for absolute value?
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Activity 2: Exploring other Absolute Value Functions
Do the following for each of the following equations:
i) Fill in the table
ii) Graph the points
iii) Draw the lines (use a ruler)
iv) Give the domain and range of the function
v) Record your observations about this new equation when
compared to the parent function
vi) Give the ordered pair of the vertex of the function
a. g(x) = |x| + 2
X
-3
-2
-1
0
1
2
3
g(x)
Domain:
Range:
Observations:
Vertex:
b. g(x) = |x| - 5
X
-3
-2
-1
0
1
2
3
g(x)
Domain:
Range:
Observations:
Vertex:
c. g(x) = |x – 3|
X
-1
0
1
2
3
4
5
6
7
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g(x)
Domain:
Range:
Observations:
Vertex:
2
d. h(x) = |x + 2|
X
-5
-4
-3
-2
-1
0
1
h(x)
Domain:
Range:
Observations:
Vertex:
e. h(x) = |x – 5| + 2
h(x)
X
2
3
4
5
6
7
8
Domain:
Range:
Observations:
Vertex:
Activity 3: Describing Graphs of Absolute Value Functions
Earlier it was mentioned that linear functions either increase at a constant rate or decrease at
a constant rate (only exception, horizontal and vertical lines). Look back at the graphs you made
in activities #1 and #2. How would you describe the functions and how they change? For
instance, I could say “some of the functions decrease at a constant rate, but then….”
You provide the rest of the details for both types of graphs while using complete sentences.
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Activity 4: Graphing without a Table
Consider the function f(x) = |x| + 4
a.
Using your results from Activity 2, how do you expect the
graph of this function to compare to the graph of the
parent function, f(x) = |x|?
b. What is the vertex for this function?
c.
Graph the function without making a table of values.
Consider the function g(x) = |x + 3| – 4
d. Using your results from Activity 2, how do you expect the
graph of this function to compare to the graph of the
parent function, f(x) = |x|?
e.
What is the vertex for this function?
f.
Graph the function without making a table of values.
g.
In general, how does the function g(x) = |x - h| + k compare
to the parent function, f(x) = |x|?
h. What is the vertex for the function g(x) = |x - h| + k?
Activity 5: Writing Equations
Write a function for each translation of the parent function f(x) = |x|.
a.
A horizontal translation of 7 units to the right
b. A vertical translation of 9 units down
c.
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A horizontal translation of 8 units to the left and a vertical translation of 10 units up
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5.8a (Day 2) Absolute Value Function Graphs
Algebra 1
Name ____________________________________
Activity 1: Absolute Value on Your Graphing Calculator
It is possible to graph absolute value functions on your calculator.
a. First clear out your “y =” equations. Place your cursor after y1 = .
b. Press the “math” button on the left side of your calculator and arrow right to “NUM”.
c. Absolute value is the first option “abs(”. Select it.
d. Type in “X” using the
key, so the equation from #1 looks like this: y1 = abs(x).
e.
Adjust your window. Check the graph shape and the location (using “trace”). Then press
2nd graph to check your table. Do they both match your answers for the parent function
f(x) = |x|?
Remember that whatever is inside the parentheses is inside the absolute value symbols.
Activity 2: Stretching/Shrinking Absolute Value Function Graphs
Do the following for each of the following equations:
i) Use your graphing calculator to create the tables and graphs for the equation
ii) Add the new graph to the graph of the parent function f(x) = |x|
a. y1 = 2|x|
x
-2
-1
0
1
2
y1
b. y2 = -|x|
x
-2
-1
0
1
2
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y2
5
c. y3 = ½ |x|
x
-2
-1
0
1
2
y3
d. y4 = 3|x|
x
-2
-1
0
1
2
y4
a) Describe the effects on the graph caused by multiplying |x| by a number.
b) Describe the effect on the graph caused by multiplying |x| by a negative number.
c) Are the domain and/or range different than the parent function’s domain and/or range for
any of the functions that you graphed in part (b)? If yes, give the function and its domain
and range.
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Activity 3: Combining Transformations
For each function:
i.
Make a prediction how you think the graph of the function will compare to the graph of
the parent function.
ii.
Use your calculator to make a graph and check your prediction.
iii.
Give the domain and range of the function.
iv.
Give the ordered pair of the vertex of the function.
a.
g(x) = 5|x – 4| + 2
Prediction:
Graph:
Domain:
Range:
Vertex:
1
b. ℎ(𝑥) = − |𝑥 + 1| − 3
3
Prediction:
Graph:
Domain:
Range:
Vertex:
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Activity 4: Changing Parameters
Absolute Value Linear Functions may be written in the form of
and k affect the parent function
y = x.
y = a x - h + k, where each of the values a, h,
In the eleven graphs you created in Activity 2, you should have
noticed how those variables (a, h and k) affected the parent function.
In your group, discuss and record what affect each of the variables has on the parent function.
Make sure your use mathematical vocabulary and be very specific.
a=
h=
k=
Activity 5: Graphing Absolute Value Functions
For the following absolute value functions, first imagine in your head the graph of the parent
function y = x . Second, decide how the values of a, h, and k will affect the parent function. Third
make a quick sketch of your prediction of the graph of the given function (do not use a calculator!).
Finally, tell what transformations occurred to the parent function to arrive at the new function.
a.
y = x +6
b.
y = x-3
c.
y = x+5 -7
d.
y=
e.
y = -2 x - 2 + 5
f.
y =-
2
3
x -6 +3
3
2
x -1 -1
Activity 7: Check Your Work
Using your calculator, check your predictions. Make any necessary adjustments.
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