# 1.6 Absolute value functions

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In this section you will:
• Graph an absolute value function.
• Solve an absolute value equation.
• Solve an absolute value inequality.

Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right, at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will investigate absolute value functions .

## Understanding absolute value

Recall that in its basic form $\text{\hspace{0.17em}}f\left(x\right)=|x|,\text{\hspace{0.17em}}$ the absolute value function, is one of our toolkit functions. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign.

## Absolute value function

The absolute value function can be defined as a piecewise function

$\text{\hspace{0.17em}}f\left(x\right)=|x|=\left\{\begin{array}{ccc}x& \text{if}& x\ge 0\\ -x& \text{if}& x<0\end{array}\text{\hspace{0.17em}}$

## Determine a number within a prescribed distance

Describe all values $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ within or including a distance of 4 from the number 5.

We want the distance between $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and 5 to be less than or equal to 4. We can draw a number line, such as the one in [link] , to represent the condition to be satisfied.

The distance from $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ to 5 can be represented using the absolute value as $\text{\hspace{0.17em}}|x-5|.\text{\hspace{0.17em}}$ We want the values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ that satisfy the condition $|x-5|\le 4.$

Describe all values $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ within a distance of 3 from the number 2.

$|x-2|\le 3$

## Resistance of a resistor

Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often $\text{\hspace{0.17em}}\text{±1%,}\text{\hspace{0.17em}}±\text{5%,}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}±\text{10%}\text{.}$

Suppose we have a resistor rated at 680 ohms, $\text{\hspace{0.17em}}±5%.\text{\hspace{0.17em}}$ Use the absolute value function to express the range of possible values of the actual resistance.

5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance $\text{\hspace{0.17em}}R\text{\hspace{0.17em}}$ in ohms,

$|R-680|\le 34$

Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation.

using the variable $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ for passing, $\text{\hspace{0.17em}}|p-80|\le 20$

## Graphing an absolute value function

The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the origin in [link] .

[link] shows the graph of $\text{\hspace{0.17em}}y=2|x–3|+4.\text{\hspace{0.17em}}$ The graph of $\text{\hspace{0.17em}}y=|x|\text{\hspace{0.17em}}$ has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at $\text{\hspace{0.17em}}\left(3,4\right)\text{\hspace{0.17em}}$ for this transformed function.

how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim