Absolute Value Function
Example
Graph the function f(x) = -|x - 3|.
Solution
| Since this is an absolute value function, we first find
the vertex. To do this, set the quantity inside the
absolute value symbols equal to 0.
|
x - 3 = 0 |
| Solve for x. |
x = 3 |
So, the x-value of the vertex is 3. Therefore, to create a table for this
function we will use x = 3 as one input value and choose x-values on
either side of 3.
We’ll let x = -2, 0, 3, 6, and 8.
Substitute those values of x into the function and simplify.
|
x |
f(x)
= - |x - 3| |
(x, y) |
|
-2
0
3
6
8 |
f(-2) = -|-2 - 3| = -5
f(0) = -|0 - 3| = -3
f(6) = -|6 - 3| = -3
f(3) = -|3 - 3| = 0
f(8)= -|8 - 3| = -5 |
(-2, -5) (0, -3)
(3, 0)
(6, -3)
(8, -5) |
Now, plot the points and connect them.
Notice that the graph of
f(x) = -|x - 3| is an inverted
 . This is because of the negative sign
in front of the absolute value.
The domain of f(x) = |x - 3| is all real numbers because we can find the
absolute value of any real number.
To find the range, examine the y-values on the graph. The y-values consist
of all real numbers less than or equal to 0.
Thus, the range is y ≤ 0. This is the interval (-∞, 0].
Notes:
In f(x) = -|x - 3|, the expression x - 3 is
linear. That is why the graph on each side
of x = 3 is a straight line.
Be careful with the negative signs when
evaluating functions like
f(x) = -|x - 3|. First, find the value of |x - 3|. Then
multiply the result by -1.
For example:
f(-4) = -|-4 -3| = -|-7| = -(7) = -7
f(3) = -|3 - 3| = -(0) = 0
|
