Graph Absolute Value Functions using Transformations

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Presentation transcript:

Graph 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 right of x = 0 the line is y = x 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.

Vocabulary 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. An absolute value graph has one axis of symmetry that passes through the vertex.

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) zero. On this graph where x = -3 and x = 3 are where the function would equal 0. f(x) = |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.

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

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

Example 1 Continued: 3. y = 2 |x + 3| - 1 4. y = -1/3|x – 2| + 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: Graph y = -2 |x + 3| + 2. What is your vertex? Vertex at (-3,2) What are the intercepts? y-intercept at y= -4 What are the zeros? X= -2 and X= -4

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

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

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