# Absolute Value Functions (2023)

An absolute value function is a function that contains an algebraic expression within absolute value symbols. Recall that the absolute value of a number is its distance from $0$ on the number line.

The absolute value parent function, written as $f\left(x\right)=|\text{\hspace{0.17em}}x\text{\hspace{0.17em}}|$, is defined as

$f\left(x\right)=\left\{\begin{array}{l}x\text{if}x>0\\ 0\text{if}x=0\\ -x\text{if}x<0\end{array}$

To graph an absolute value function, choose several values of $x$ and find some ordered pairs.

 $x$ $y=|\text{\hspace{0.17em}}x\text{\hspace{0.17em}}|$ −2 2 −1 1 0 0 1 1 2 2

Plot the points on a coordinate plane and connect them. Observe that the graph is V-shaped.

($1$) The vertex of the graph is $\left(0,0\right)$.

($2$) The axis of symmetry ($x=0$ or $y$-axis) is the line that divides the graph into two congruent halves.

($3$) The domain is the set of all real numbers.

($4$) The range is the set of all real numbers greater than or equal to $0$. That is, $y\ge 0$.

($5$) The $x$-intercept and the $y$-intercept are both $0$.

### Vertical Shift

To translate the absolute value function $f\left(x\right)=|\text{\hspace{0.17em}}x\text{\hspace{0.17em}}|$ vertically, you can use the function

$g\left(x\right)=f\left(x\right)+k$.

When $k>0$, the graph of $g\left(x\right)$ translated $k$ units up. When $k<0$, the graph of $g\left(x\right)$ translated $k$ units down. ### Horizontal Shift

To translate the absolute value function $f\left(x\right)=|\text{\hspace{0.17em}}x\text{\hspace{0.17em}}|$ horizontally, you can use the function

$g\left(x\right)=f\left(x-h\right)$.

When $h>0$, the graph of $f\left(x\right)$ is translated $h$ units to the right to get $g\left(x\right)$. When $h<0$, the graph of $f\left(x\right)$ is translated $h$ units to the left to get $g\left(x\right)$. ## Stretch and Compression

The stretching or compressing of the absolute value function $y=|\text{\hspace{0.17em}}x\text{\hspace{0.17em}}|$is defined by the function $y=a|\text{\hspace{0.17em}}x\text{\hspace{0.17em}}|$where $a$ is a constant. The graph opens up if $a>0$and opens down when $a<0$. For absolute value equations multiplied by a constant $\left(\text{for example,}y=a|\text{\hspace{0.17em}}x\text{\hspace{0.17em}}|\right)$,if $0, then the graph is compressed, and if $a>1$, it is stretched. Also, if a is negative, then the graph opens downward, instead of upwards as usual. More generally, the form of the equation for an absolute value function is $y=a|\text{\hspace{0.17em}}x-h\text{\hspace{0.17em}}|+k$. Also:

• The vertex of the graph is $\left(h,k\right)$.
• The domain of the graph is set of all real numbers and the range is $y\ge k$when $a>0$.
• The domain of the graph is set of all real numbers and the range is $y\le k$when $a<0$.
• The axis of symmetry is $x=h$.
• It opens up if $a>0$and opens down if $a<0$.
• The graph $y=|\text{\hspace{0.17em}}x\text{\hspace{0.17em}}|$can be translated $h$ units horizontally and $k$ units vertically to get the graph of $y=a|\text{\hspace{0.17em}}x-h\text{\hspace{0.17em}}|+k$.
• The graph $y=a|\text{\hspace{0.17em}}x\text{\hspace{0.17em}}|$is wider than the graph of $y=|\text{\hspace{0.17em}}x\text{\hspace{0.17em}}|$ if $|\text{\hspace{0.17em}}a\text{\hspace{0.17em}}|<1$and narrower if $|a|>1$.

## FAQs

### How many answers should you get for an absolute value problem? ›

An absolute value equation may have one solution, two solutions, or no solutions.

Do you have to check your answers for absolute value equations? ›

Second, if the absolute value equation is of the form , we don't need to check for any solutions at all. This is because an absolute value expression can never be negative, so there will never be a solution for an equation of this form.

Do all absolute value equations have 2 answers? ›

In conclusion, an absolute-value problem will not always have two solutions, because absolute-value inequalities result in one set of solutions. If the problem contains only one absolute number, it will have only one solution and that will be the positive number.

How many answers do you usually have when solving an absolute value equation? ›

Summary. Absolute value equations are always solved with the same steps: isolate the absolute value term and then write equations based on the definition of the absolute value. There may end up being two solutions, one solution, or no solutions.

Why are there two answers to absolute value equations? ›

And represents the distance between a and 0 on a number line. Has two solutions x = a and x = -a because both numbers are at the distance a from 0.

What are the rules of absolute value functions? ›

For absolute value equations multiplied by a constant (for example,y=a| x |),if 0<a<1, then the graph is compressed, and if a>1, it is stretched. Also, if a is negative, then the graph opens downward, instead of upwards as usual. More generally, the form of the equation for an absolute value function is y=a| x−h |+k.

What is the rule for absolute value equation? ›

Definitions: The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign. For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5.

What is an absolute value equation with only one solution? ›

For example, |x|=0 has only one solution x=0 .

Can absolute value equations have infinite solutions? ›

Yes, it can; for example, |x|=x has the solution set [0,∞) , which contains infinitely many solutions.

How many solutions can an absolute value inequality have? ›

Absolute value equations can have up to two solutions. To solve an absolute value inequality involving “less than,” such as |X|≤p, replace it with the compound inequality −p≤X≤p and then solve as usual.

### What is common for the equations that have no solution? ›

A system of linear equations has no solution when the graphs are parallel.

Which equation has no solution? ›

The last type of equation is known as a contradiction, which is also known as a No Solution Equation. This type of equation is never true, no matter what we replace the variable with. As an example, consider 3x + 5 = 3x - 5. This equation has no solution.

What makes an absolute value equation false? ›

There's one MAJOR red flag of an equation with an absolute value that has no solution. Recall that an absolute value expression can never be less than zero. That is, a fully reduced absolute value expression must be greater than or equal to zero.

Why is there no solution to an absolute value equation when the absolute value is equal to a negative number? ›

Since the absolute value of any number other than zero is positive, it is not permissible to set an absolute value expression equal to a negative number. So, if your absolute value expression is set equal to a negative number, then you will have no solution.

What are the 4 steps to solving an absolute value equation? ›

Step 1: Isolate the absolute value|3x - 6| - 9 = -3 |3x - 6| = 6
Step 2: Is the number on the other side of the equation negative?No, it's a positive number, 6, so continue on to step 3
Step 3: Write two equations without absolute value bars3x - 6 = 6
Step 4: Solve both equations3x - 6 = 6 3x = 12 x = 4

How can you tell whether an absolute value function has two without graphing the function? ›

So, to determine whether an absolute value function has two x-intercepts without graphing, check the value of c. If c < 0, then the function has two x-intercepts. If c ≥ 0, then the function has either one or no x-intercepts.

Is there a limit for absolute value function? ›

Limits involving absolute values often involve breaking things into cases. Remember that |f(x)|={f(x), if f(x)≥0;−f(x), if f(x)≤0. By studying these cases separately, we can often get a good picture of what a function is doing just to the left of x=a, and just to the right of x=a.

What are the 4 properties of absolute value? ›

Absolute value has the following fundamental properties: Non-negativity |a| ≥ 0. Positive-definiteness |a| = 0a = 0. Multiplicativity |ab| = |a| |b|

How many numbers can have the same absolute value? ›

Answer: Two different integers can have the same absolute value.

Can there be two absolute values? ›

Yes, but only if there are exactly just the two absolute values, so that we can "isolate" each of them, one on either side of the equation.

### Are all absolute value functions one to one? ›

As with onto, whether a function is one-to-one frequently depends on its type signature. For example, the absolute value function |x| is not one-to- one as a function from the reals to the reals. However, it is one-to-one as a function from the natural numbers to the natural numbers.

What is an example of an infinite solution? ›

An infinite solution has both sides equal. For example, 6x + 2y - 8 = 12x +4y - 16. If you simplify the equation using an infinite solutions formula or method, you'll get both sides equal, hence, it is an infinite solution.

How do you tell if a system has infinite solutions or no solutions? ›

Conditions for Infinite Solution

If the two lines have the same y-intercept and the slope, they are actually in the same exact line. In other words, when the two lines are the same line, then the system should have infinite solutions.

What equation have infinitely many solutions? ›

The equation 2x + 3 = x + x + 3 is an example of an equation that has an infinite number of solutions.

Can an absolute value be infinity? ›

The Absolute Infinite (symbol: Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor. It can be thought of as a number that is bigger than any other conceivable or inconceivable quantity, either finite or transfinite.

What is the absolute value of 4 responses? ›

Answer and Explanation: The absolute value of 4 is 4. The formal definition of the absolute value of a number x, denoted |x|, is the distance that x is from 0 on a number line. Since distance is always positive, we see that taking the absolute value of a number is the same as taking the positive value of that number.

Why there is more than 1 answer when working with absolute value problems? ›

Because two numbers have the same absolute value (except 0 ).

What is the rule for absolute value? ›

The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign. For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5. The absolute value of a number may be thought of as its distance from zero along real number line.

What is the absolute value of − 6 responses 6? ›

Absolute Value Examples and Equations

|–6| = 6 means “the absolute value of –6 is 6.”

What is the absolute value of 9 responses? ›

The absolute value of a number tells only one thing: 1. distance from 0. The absolute value of 9 is 9. (9 is 9 places from 0.)

### Why the absolute value of 4 |- 4 is 4? ›

The absolute value of a number is its magnitude, or distance from 0. We write "the absolute value of 4" as |4|, and |4| is 4, since it's 4 units away from 0 on a number line.

Does absolute value have a limit? ›

Limits involving absolute values often involve breaking things into cases. Remember that |f(x)|={f(x), if f(x)≥0;−f(x), if f(x)≤0. By studying these cases separately, we can often get a good picture of what a function is doing just to the left of x=a, and just to the right of x=a.

Why is absolute value always positive? ›

Absolute Value Symbol

The distance of any number from the origin on the number line is the absolute value of that number. It also shows the polarity of the number whether it is positive or negative. It can be negative ever a s it shows the distance and the distance can't be negative. So, it is always positive.

Can absolute value have infinite solutions? ›

Yes, it can; for example, |x|=x has the solution set [0,∞) , which contains infinitely many solutions.

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