# 3.6 Absolute value functions  (Page 2/3)

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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 $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(x\right).$

$f\left(x\right)=a|x-3|-2$

Now substituting in the point (1, 2)

$\begin{array}{ccc}\hfill 2& =& a|1-3|-2\hfill \\ \hfill 4& =& 2a\hfill \\ \hfill a& =& 2\hfill \end{array}$

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

$f\left(x\right)=-|x+2|+3$

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] ).

## 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 $\text{\hspace{0.17em}}8=|2x-6|,\text{\hspace{0.17em}}$ 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.

$\begin{array}{ccccccc}\hfill 2x-6& =& 8\hfill & \phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}& \hfill 2x-6& =& -8\hfill \\ \hfill 2x& =& 14\hfill & & \hfill 2x& =& -2\hfill \\ \hfill x& =& 7\hfill & & \hfill x& =& -1\hfill \end{array}$

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,

$\begin{array}{l}|x|=4,\hfill \\ |2x-1|=3,\text{or}\hfill \\ |5x+2|-4=9\hfill \end{array}$

## Solutions to absolute value equations

For real numbers $A$ and $B$ , an equation of the form $|A|=B,$ with $B\ge 0,$ 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 $\text{\hspace{0.17em}}|A|=B\text{\hspace{0.17em}}$ to write $\text{\hspace{0.17em}}A=B\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\mathrm{-A}=B,\text{\hspace{0.17em}}$ assuming $\text{\hspace{0.17em}}B>0.$
3. Solve for $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$

## Finding the zeros of an absolute value function

For the function $\text{\hspace{0.17em}}f\left(x\right)=|4x+1|-7,$ find the values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}f\left(x\right)=0.$

The function outputs 0 when $\text{\hspace{0.17em}}x=\frac{3}{2}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}x=-2.$ See [link] .

For the function $\text{\hspace{0.17em}}f\left(x\right)=|2x-1|-3,$ find the values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}f\left(x\right)=0.$

$x=-1\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=2$

Should we always expect two answers when solving $\text{\hspace{0.17em}}|A|=B?$

No. We may find one, two, or even no answers. For example, there is no solution to $\text{\hspace{0.17em}}2+|3x-5|=1.$

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
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?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
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
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
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.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
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I rally confuse this number And equations too I need exactly help
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salma
Commplementary angles
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