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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

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
what is the coefficient of -4×
Mehri Reply
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
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I'm 13 and I understand it great
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
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is it a question of log
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Commplementary angles
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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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