TRANSFORMING ABSOLUTE VALUE FUNCTIONS

The parent function for absolute-value functions is

f(x)  =  |x|

The graphs of all other absolute-value functions are transformations of the graph of f(x) = |x|.

We can consider the following transformations for 

f(x)  =  a|x - b| + c

Opening Direction

• If a > 0, the graph opens upward.

• If a < 0, the graph opens downward.

Width

• If |a| > 1, the graph is narrower than the graph of f(x) = |x| .

• If |a| < 1, the graph is wider than the graph of f(x) = |x|.

The slopes of the two linear pieces are a and –a.

Horizontal Translation

• If b > 0, the graph is translated b units right from f(x) = |x|.

• If b < 0, the graph is translated b units left from f(x) = |x|.

Vertical Translation

• If c > 0, the graph is translated c units up from f(x) = |x|.

• If c < 0, the graph is translated c units down from f(x) = |x|.

Solved Examples

Describe the transformations from the graph of f(x) = |x| to the graph of g(x). Then graph both functions.

Example 1 :

g(x)  =  2|x + 1|

Solution :

Identify a, b, and c.

g(x) = 2|x + 1| = 2|x – (–1)| + 0.

• a = 2 : graph is narrower

• b = –1 : translated 1 unit left

• c = 0 : no vertical translation

Example 2 : 

g (x)  =  -|x - 3| + 2

Solution :

Identify a, b, and c.

g(x) = -|x - 3| + 2 = -1|x – 3| + 2.

• a = -1 : graph opens downward and width is unchanged

• b = 3 : translated 3 units right

• c = 2 : translated 2 units up

Sports Application

Example 3 :

In a charity race, a water stand for the runners is halfway between the start and finish lines. The absolute value function y = |(x/8) - 3|models Riley’s distance y in miles from the water stand x minutes into the race. The function y = |(x/10) - 3| models Dean’s distance from the water stand during the same race. Compare Dean’s graph to Riley’s graph. What can you conclude about Dean’s speed?

Solution :

y = |(x/8) - 3| is graphed in blue. 

y = |(x/10) - 3| is graphed in red. Both graphs start at the same point, but Dean’s graph is translated to the right. It takes him more time to reach the water stand and to finish the race. Therefore, he is running more slowly than Riley.

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