Graph Absolute Value Functions using Transformations Vocabulary l

Graph Absolute Value Functions using Transformations

Vocabulary l 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 Notice that the graph is symmetric over the y-axis because for every point (x, y) on the graph, the point (-x, y) is also on it. To the right of x = 0 the line is y=x

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

l Absolute Value Function l Vertex l Axis of Symmetry

Vocabulary l The zeros of a function f(x) are the values of x that make the value of f(x) zero. l On this graph where x = -3 and x = 3 are where the function would equal 0. f(x) = |x| - 3

Vocabulary l l l 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(-1 x).

Vocabulary l 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 Reflection across the x-axis Vertical Stretch a>1 (makes it narrower) OR Vertical Compression 0<a<1 (makes it wider) Vertical Translation Horizontal Translation (opposite of h) *Remember that (h, k) is your vertex*

Example 1: l Identify the transformations: 1. y = 3 |x + 2| - 3 • • • Vertically stretched by a factor of 3 Shifted left two units Shifted down three units y = |x – 1| + 2 2. • • Shifted right one unit Shifted up two units

Example 1 Continued: 3. y = 2 |x + 3| - 1 • • • Vertically stretched by a factor of 2 Shifted left three units Shifted down one unit 4. y = -1/3|x – 2| + 1 • • Shifted right two units Shifted up one unit Vertically compressed by a factor of 1/3 Reflected over the x-axis

Example 2: l Graph y = -2 |x + 3| + 2. l What is your vertex? l l What are the intercepts? l l Vertex at (-3, 2) y-intercept at y= -4 What are the zeros? l X= -2 and X= -4

You Try: l Graph y = -1/2 |x – 1| - 2 Compare the graph with the graph of y = |x| (what are the transformations) l • • Shifted down two units Shifted right one unit Vertically Compressed by a factor of ½ Reflected over the x-axis

Example 3: l Write a function for the graph shown. y= -2|x-3|+2

You Try: l Write a function for the graph shown. y= 2|x+1|+3