Absolute value functions and transformations.notebook

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Absolute Value Functions and Graphs

Objective:I can graph an absolute value function by performing transformations on the parent function f (x) = |x|.

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

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Absolute Value Function

· Absolute Value Function

· Vertex

· Axis of Symmetry

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Building the Absolute Value FunctionThe absolute value function is defined by f (x) = |x|.

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This is the absolute value parent function.Table

x yf(x)=lxl

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Parent Function

· V­shape· It is symmetric about the y­axis (Axis of Symmetry)· The vertex is the minimum point on the graph

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TranslationA translation is a transformation

that shifts a graph horizontally or vertically, but doesn’t change the overall shape or orientation.

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Stretching and CompressionThe graph of y = a|x| is graph of y = |x| vertically stretched or compressed depending on the |a|.

The value of a acts like the slope.

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ReflectionThe graph of y = a|x| is graph of y = |x| reflected across the x­axis.

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y = ­a |x – h| + k

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Multiple TransformationsIn general, the graph of an absolute value function of the form y = a|x – h| + k can involve translations, reflections, stretches or compressions.

To graph an absolute value function, start by identifying the vertex

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Graphing Absolute Value FunctionsGraphing y = a|x – h| + k is straight forward:

1. Plot the vertex (h, k). (note…if +h inside that means h is negative(to the left); if – h inside that means h is positive (to the right)

2. Use the a value as slope to plot more points. Remember you have to do positive and negative slope to get points on both sides of the V

3. Connect the dots in a V­shape.

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Example 1Graph the following function without making a table.

y = |x – 2| + 3

Vertex

How does this graph move?

Did this graph stretch or compress?

(slope of 1)

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Example 2Graph the following function.

y = (1/2)|x|

Vertex

How does this graph move?

Did this graph stretch or compress?

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Your turn:

y = (3/2)|x|

Vertex ( , )Translations

Slope

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y = 2|x + 2| ­ 3

Your turn:

Vertex ( , )Translations

Slope

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write an abs value function that...