Graphing Absolute Value Functions using Transformations

Vocabulary The function f(x) = |x| is an absolute value function.

The graph of this piecewise function consists of 2 rays, is v-shaped, and opens up. to the left of x = 0 the line is y = -x to the right of x = 0 the line is y = x Notice that the graph is symmetric across the y-axis because for every point (x,y) on the graph, the point (-x,y) is also on the graph.

Vocabulary The highest or lowest point on the graph of an absolute value function is called the vertex. The axis of symmetry of the graph of a function is a vertical line that divides the graph into mirror images.

Absolute Value Function Vertex Axis of Symmetry

Vocabulary The zeros of a function f(x) are the values of x that make the value of f(x) equal to 0. f(x) = |x| - 3 On this graph, f(x) (or y) is 0 when x = -3 and x = 3.

Vocabulary A transformation changes a graph’s size, shape, position, or orientation. A translation is a transformation that shifts a graph horizontally and/or vertically, but does not change its size, shape, or orientation. A reflection is when a graph is flipped over a line. A graph flips vertically when -1. f(x) and it flips horizontally when f(-1x).

Vocabulary A dilation changes the size of a graph by stretching or compressing it. This happens when you multiply the function by a number.

Transformations y = -a |x – h| + k remember that (h, k) is your vertex reflection across the x-axis vertical stretch a > 1 (makes it narrower) OR vertical compression 0 < a < 1 (makes it wider) horizontal translation (opposite of h) vertical translation

Example 1 Identify the transformations. 1. y = 3 |x + 2| y = |x – 1| y = 2 |x + 3| y = -1/3|x – 2| + 1

Example 2 Graph y = -2 |x + 3| + 2. What is the vertex? What are the intercepts?

You Try: Graph y = -1/2 |x – 1| - 2  compare the graph with the graph of y = |x| What are the transformations?

Example 3 Write a function for the graph shown.

You Try: Write a function for the graph shown.