2.6 families of functions - warren hills regional school ...€¦ · families of functions...
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FAMILIES OF FUNCTIONS
Objectives: 1. To graph an absolute value function by
performing transformations on the parent
2. To apply transformations to graphing any function
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Building the Absolute Value Function
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 Function
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 Function
The absolute value function is defined by f (x) = |x|.
This is the absolute value parent function.
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f x( ) = x
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Parent Function
• V-shape • It is symmetric about the y-axis • The vertex is the minimum point on the graph
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Translation
A translation is a transformation that shifts a graph horizontally or vertically, but doesn’t change the overall shape or orientation.
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Translation
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)
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Stretching and Shrinking The graph of y = a|x| is graph of y = |x| vertically stretched
or shrunk depending on the |a|.
The value of a acts like the slope.
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Reflection The graph of y = a|x| is graph of y = |x| reflected across the
x-axis when a < 0.
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f x( ) = x
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f x( ) = - x
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Multiple Transformations
In general, the graph of an absolute value function of the form y = a|x – h| + k can involve translations, reflections, stretches or shrinks.
To graph an absolute value function, start by identifying the vertex.
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Graphing Absolute Value Functions
Graphing y = a|x – h| + k is easy: 1. Plot the vertex (h, k). (note…if +h inside that means h is negtive, if – h inside that means h is positive)
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 V-shape.
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Exercise
Graph the following functions without making a table. 1. 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
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Exercise Graph the following functions without making a table. 1. 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
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Exercise Graph the following functions without making a table. 1. 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
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Transformations in General You can perform transformations on the graph of any function in manner similar to transformations on the absolute value function.