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# 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.

Weâ€™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

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