Definition: The absolute value function f(x) = |x| returns the distance of x from zero, and its graph is a V shape with its vertex at the origin.
What does the graph look like?
For x at least 0, |x| = x, so the right arm is the line y = x. For x below 0, |x| = -x, so the left arm is the line y = -x. The two arms meet at the vertex (0, 0) and the output is never negative.
How do transformations work?
In y = a|x – h| + k, the vertex is (h, k). A negative a flips the V upside down, and a larger |a| makes it narrower.
Example. Graph y = 2|x + 3| – 5. The vertex is (-3, -5). One step right of the vertex, x = -2, gives y = 2(1) – 5 = -3. One step left, x = -4, gives -3 as well. The domain is all real numbers, and the range is y ≥ -5.
Which errors are typical?
Writing the vertex as (3, -5) because of the sign inside, and taking |-7| as -7. Solving |x – 4| = 9 also needs two cases: x – 4 = 9 or x – 4 = -9, so x = 13 or x = -5. The graph is symmetric about the vertical line through its vertex.
Where to learn it
- Key Features of Graphs and Rates of Change – lesson
- Shifting, Stretching and Piecewise Functions – lesson