2.7: Absolute Value Functions and 2.7: Absolute Value Functions and GraphsGraphs

Objective:

SWBAT graph an absolute value function by performing transformations on the parent function f (x) = |x|.

VocabularyVocabulary

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

• The highest of 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

Absolute Value Function

Building the Absolute Value Building the Absolute Value FunctionFunction

The absolute value function is defined by f (x) = |x|.

The graph of the absolute value function is similar to the linear parent function, except it must always be positive.

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Building the Absolute Value Building the Absolute Value FunctionFunction

The absolute value function is defined by f (x) = |x|.

So we just take the negative portion of the graph and reflect it across the x-axis making that part positive.

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Building the Absolute Value Building the Absolute Value FunctionFunction

The absolute value function is defined by f (x) = |x|.

This is the absolute value parent functionparent function.

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f x = x

Parent FunctionParent Function

• V-shape• It is symmetric about the y-axis (Axis of

Symmetry)• The vertexvertex is the minimum point on the graph

TranslationTranslation

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

TranslationTranslation

The graph of

y = |x – h| + k

is the graph of y = |x| translated h horizontal units and y vertical units.

• The new vertex is at (h, k)

Stretching and CompressionStretching and Compression

The 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.

ReflectionReflection

The graph of y = a|x| is graph of y = |x| reflected across the x-axis.

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f x = x

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f x = - x

TransformationsTransformations

y = -a |x – h| + k

*Remember that (h, k) is your vertex*

Reflection across the

x-axis Vertical Stretcha > 1

(makes it narrower)

ORVertical

Compression 0 < a < 1

(makes it wider)

Horizontal Translation

(opposite of h)

Vertical Translation

Multiple TransformationsMultiple Transformations

In 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.

Graphing Absolute Value FunctionsGraphing Absolute Value Functions

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

• 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)

• 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

• Connect the dots in a V-shape.

Example 1Example 1

Graph the following functions without making a table.

y = |x – 2| + 3 This graph will go right 2 and up 3 so from the origin go right 2 and up 3. This is the vertex (2, 3). Now from that point use the positive and negative slope (a = 1 here) to get more points.

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f x = x

Your turn:Your turn:

Graph the following functions without making a table.

Text book page 111 #14

y = |x - 1| + 3

Identify the vertex

Vertex ( , )

Slope =

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f x = x

Example 2Example 2

Graph the following functions without making a table.

y = (1/2)|x| This function does not have an “h” or “k” so the vertex is (0, 0). Since a = ½ the slope is ½. Go up 1 and right 2 then up one and left 2.

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f x = x

Your turn:Your turn:

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f x = x

Example 3Example 3

Graph the following functions without making a table.

f (x) = -3|x + 1| – 2 This graph will go left 1 and down two so the vertex will be (-1, -2). Since “a” is negative the graph will open down. Since the value of “a” is 3 the slope will be 3 and -3 (just remember to go down.)

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f x = x

Your turn:Your turn:

Graph the following functions without making a table.

Text book page 112 #42

y = 2|x + 2| - 3

Identify the vertex

Vertex ( , )

Slope =

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f x = x

HOMEWORKHOMEWORK

Don’t forget to identify VERTEX first before graphing

Homework: page 111 #17-27 (odds)

Extra Challenge question

Text book page 111 #26