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If we couldn’t observe the stretch of the function from the graphs, could we algebraically determine it?

Yes. If we are unable to determine the stretch based on the width of the graph, we can solve for the stretch factor by putting in a known pair of values for x and f ( x ) .

f ( x ) = a | x 3 | 2

Now substituting in the point (1, 2)

2 = a | 1 3 | 2 4 = 2 a a = 2

Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically flipped, and vertically shifted up 3 units.

f ( x ) = | x + 2 | + 3

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Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?

Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.

No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (see [link] ).

Graph of the different types of transformations for an absolute function.
(a) The absolute value function does not intersect the horizontal axis. (b) The absolute value function intersects the horizontal axis at one point. (c) The absolute value function intersects the horizontal axis at two points.

Solving an absolute value equation

In Other Type of Equations , we touched on the concepts of absolute value equations. Now that we understand a little more about their graphs, we can take another look at these types of equations. Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as 8 = | 2 x 6 | , we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.

2 x 6 = 8 or 2 x 6 = −8 2 x = 14 2 x = −2 x = 7 x = −1

Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.

An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,

| x | = 4 , | 2 x 1 | = 3 , or | 5 x + 2 | 4 = 9

Solutions to absolute value equations

For real numbers A and B , an equation of the form | A | = B , with B 0 , will have solutions when A = B or A = B . If B < 0 , the equation | A | = B has no solution.

Given the formula for an absolute value function, find the horizontal intercepts of its graph .

  1. Isolate the absolute value term.
  2. Use | A | = B to write A = B or −A = B , assuming B > 0.
  3. Solve for x .

Finding the zeros of an absolute value function

For the function f ( x ) = | 4 x + 1 | 7 , find the values of x such that f ( x ) = 0.

0 = | 4 x + 1 | 7 Substitute 0 for  f ( x ) . 7 = | 4 x + 1 | Isolate the absolute value on one side of the equation . 7 = 4 x + 1 or −7 = 4 x + 1 Break into two separate equations and solve . 6 = 4 x −8 = 4 x x = 6 4 = 1.5 x = 8 4 = 2

The function outputs 0 when x = 3 2 or x = 2. See [link] .

Graph an absolute function with x-intercepts at -2 and 1.5.
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For the function f ( x ) = | 2 x 1 | 3 , find the values of x such that f ( x ) = 0.

x = 1 or x = 2

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Should we always expect two answers when solving | A | = B ?

No. We may find one, two, or even no answers. For example, there is no solution to 2 + | 3 x 5 | = 1.

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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