Algebra 1
Absolute Value Functions: Graphing
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.
1. Let’s start with the simplest absolute value function: y = |x|. This equation, y = |x|, is
called the “parent function” of absolute value functions as all other absolute value
functions come from it, or build from it. Confused? It is okay for now, you will realize
what this means by the end of this investigation.
Recall that absolute values are distance from 0, and therefore always positive.
a. Fill in the following table for y = |x|.
b. Graph the resulting ordered pairs.
X
-4
-3
-2
-1
0
1
2
3
4
Y
c. 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?
f. The Domain of a function is the set of independent (x) values that work for that function. The
domain for y = |x| is “all real numbers” because any number you can think of can be put into
this equation for x. Or, you could write the domain as a compound inequality: -∞ < x < +∞.
g. The Range of a function is the set of dependent (y) values that work for that function. “All
real numbers” is not the range for y = |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. If you have not already, add the definitions of Domain and Range to your toolkit.
2. Let’s explore some 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. y = |x| + 2
X
-3
-2
-1
0
1
2
3
Y
Domain:
Range:
Comments:
Vertex:
b. y = |x| - 5
X
-3
-2
-1
0
1
2
3
Y
Domain:
Range:
Comments:
Vertex:
c. y = |x – 3|
X
-1
0
1
2
3
4
5
6
7
Y
Domain:
Range:
Comments:
Vertex:
d. y = |x + 2|
X
-5
-4
-3
-2
-1
0
1
Y
Domain:
Range:
Comments:
Vertex:
e. y = |x – 5| + 2
X
2
3
4
5
6
7
8
Y
Domain:
Range:
Comments:
Vertex:
f. y = -|x|
X
-3
-2
-1
0
1
2
3
Y
Domain:
Range:
Comments:
Vertex:
g. y = -|x| + 4
X
-3
-2
-1
0
1
2
3
Y
Domain:
Range:
Comments:
Vertex:
h. y = -|x – 2|
X
-1
0
1
2
3
4
5
Y
Domain:
Range:
Comments:
Vertex:
i. y = -|x – 3| + 6
X
0
1
2
3
4
5
6
Y
Domain:
Range:
Comments:
Vertex:
3. 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. The equation from #1 looks like this: y1 = abs(x). Then press “zoom” “6” for a standard
window. Check the graph shape and location. Then press 2nd graph to check your table.
e. Remember that whatever is inside the parentheses is inside the absolute value symbols.
f. Divide up the work in your groups and check all graphs and tables from #2. Place a
checkmark beside each one that is correct.
4. 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 10 graphs you made in problems #1 and #2. How would you describe the functions
and how they change? Complete the following two sentences:
Some of the functions decrease at a constant rate, but then…
Some of the functions increase at a constant rate, but then…
5. Absolute Value Linear Functions may be written in the form of y = a x - h + k , where each
of the values a, h, and k affect the parent function y = x differently. In the eleven graphs
you created in problem #2, you should have noticed how those variables (a, h and k) affected
the parent function.
In your group, discuss what affect each of the variables has on the parent function.
Make sure your use mathematical vocabulary and be very specific.
a=
h=
k=