graphing absolute value functions using transformations
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Graphing Absolute Value Functions using Transformations
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Vocabulary
The function f(x) = |x| is an absolute value function.
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The graph of this piecewise function consists of 2 rays, is v-shaped, and opens up.
to the left ofx = 0 the line isy = -x
to the right of x = 0 the line is y = x
Notice that the graph is symmetric across the y-axis because for every point (x,y) on the graph, the point (-x,y) is also on the graph.
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Vocabulary The highest or lowest point on the graph of an
absolute value function is called the vertex.
The axis of symmetry of the graph of a function is a vertical line that divides the graph into mirror images.
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Absolute Value Function
Vertex
Axis of Symmetry
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Vocabulary
The zeros of a function f(x) are the values of x that make the value of f(x) equal to 0.
f(x) = |x| - 3
On this graph, f(x) (or y) is 0 when x = -3 and x = 3.
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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).
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Vocabulary
A dilation changes the size of a graph by stretching or compressing it. This happens when you multiply the function by a number.
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Transformations
y = -a |x – h| + k
remember that (h, k) is your vertex
reflection across the
x-axisvertical stretch
a > 1(makes it narrower)
ORvertical compression
0 < a < 1 (makes it wider)
horizontal translation
(opposite of h)
vertical translation
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Example 1 Identify the transformations.1. y = 3 |x + 2| - 3
2. y = |x – 1| + 2
3. y = 2 |x + 3| - 1
4. y = -1/3|x – 2| + 1
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Example 2 Graph y = -2 |x + 3| + 2.
What is the vertex? What are the intercepts?
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You Try: Graph y = -1/2 |x – 1| - 2
compare the graph with the graph of y = |x|
What are the transformations?
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Example 3
Write a function for the graph shown.
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You Try:
Write a function for the graph shown.