Finding the range of absolute value functions?

Just like finding the range of a parabola, the range of an absolute value function requires

1) the coordinates of the vertex, and

2) determining which way the function "opens".

For example, a parabola such as y = (x + 2)^2 + 5 would have a vertex at (-2, 5), and the y-coordinate determines an end point for the range, depending on the value in front of the squared binomial. Generally speaking, the parabola

y = a(x - h)^2 + k

Would have a range of either

1) (-infinity, k] (if a is less than 0)

2) [k, infinity) (if a is greater than 0)

The same concept applies to absolute value. An absolute value function of the form

y = a|x - h| + k

Would have a range of either (-infinity, k] if a is less than 0, or [k, infinity) if a is greater than 0.

Let's apply this to each of the questions.

1) y = |x + 4|

This is the same thing as saying

y = (1) |x + 4| + 0

Which means our "k" value is 0. Since the number in front of the |x + 4| is positive, our range is

[0, infinity)

Which is another way of saying all values greater than or equal to 0.

2) y = |x| + 2

This is the same as saying

y = (1)|x - 0| + 2

Range: [2, infinity)

3) y = -|x + 4|

This is the same thing as saying

y = (-1) |x + 4| + 0

Notice this time that the number in front of |x + 4| is negative. That means our range is instead going to be (-infinity, 0]

4) y = -|x| + 2

This is the same as saying

y = (-1)|x - 0| + 2

Showing that our range is

(-infinity, 2]

Key things to know:

1) The constant term multiplied to the absolute value determines whether the absolute value function opens up or down, and

2) The constant term added on to the absolute value determines one of the endpoints