## Absolute Value Functions

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 00 on the number line.

The absolute value parent function, written as f(x)=| x |, is defined as

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

Observe that the graph is V-shaped.

(1) The vertex of the graph is (0,0).

(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≥0.

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

### Vertical Shift

To translate the absolute value function f(x)=| x | vertically, you can use the function

g(x)=f(x)+k.

When k>0, the graph of g(x) translated k units up.

When k<0, the graph of g(x) translated k units down.

### Horizontal Shift

To translate the absolute value function f(x)=| x | horizontally, you can use the function

g(x)=f(x−h).

When h>0, the graph of f(x) is translated h units to the right to get g(x).

When h<0, the graph of f(x) is translated h units to the left to get g(x).

## Stretch and Compression

The stretching or compressing of the absolute value function y=| x | is defined by the function y=a| x | where aa is a constant. The graph opens up if a>0 and opens down when a<0.

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. Also:

- The vertex of the graph is (h,k).
- The domain of the graph is set of all real numbers and the range is y≥k when a>0.
- The domain of the graph is set of all real numbers and the range is y≤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=| x | can be translated hh units horizontally and kk units vertically to get the graph of y=a| x−h |+k.
- The graph y=a| x | is wider than the graph of y=| x | if | a |<1 and narrower if |a|>1.

## Absolute value graphs review

The general form of an absolute value function is f(x)=a|x-h|+k. From this form, we can draw graphs. This article reviews how to draw the graphs of absolute value functions.

General form of an absolute value equation:

*f*(*x*)=*a*∣*x*−*h*∣+*k*

The variable *a* tells us how far the graph stretches vertically, and whether the graph opens up or down. The variables *h* and k tell us how far the graph shifts horizontally and vertically.

Some examples:

### Example problem 1

We’re asked to graph:

*f*(*x*)=∣*x*−1∣+5

First, let’s compare with the general form:

*f*(*x*)=*a*∣*x*−*h*∣+*k*

The value of *a* is 1, so the graph opens upwards with a slope of 1 (to the right of the vertex).

The value of h is 1 and the value of k is 5, so the vertex of the graph is shifted 1 to the right and 5 up from the origin.

Finally here’s the graph of y=f(x):

### Example problem 2

We’re asked to graph:

*f*(*x*)=−2∣*x*∣+4

First, let’s compare with the general form:

*f*(*x*)=*a*∣*x*−*h*∣+*k*

The value of a is -2, so the graph opens downwards with a slope of -2 (to the right of the vertex).

The value of his 0 and the value of k is 4, so the vertex of the graph is shifted 4 up from the origin.

Finally here’s the graph of *y*=*f*(*x*):

### Absolute Value

The absolute value of a variable x is represented by |x| which is pronounced as ‘Mod x’ or ‘Modulus of x’. ‘Modulus’ is a Latin word, which means ‘measure’. Absolute value is commonly referred to as numeric value or magnitude. The absolute value represents only the numeric value and does not include the sign of the numeric value. The modulus of any vector quantity is always taken positive and is its absolute value. Also, quantities like distance, price, volume, time, are always represented as absolute values.

As an example the absolute value: |+5| = |-5| = 5. There is no sign assigned to absolute value.

**What are Absolute Values?**

The absolute value of a number is its distance from 0. We know that distance is always a non-negative quantity. Since the absolute value is a distance, the absolute value is always non-negative. Sometimes a sign is attributed to a numeric value to signify the direction, in addition to the value. The increase or a decrease of a quantity, values above or below the mean value, profit, or loss in a transaction, is sometimes explained by assigning a positive or negative value to the numeric value. But for absolute value, the sign of the numeric value is ignored and only the numeric value is considered.

In the above figure we can observe the absolute values on the number line using the illustration. The absolute value is represented by |x|, and in the above illustration, | 4 | = | -4 | = 4.

**What is the Absolute Value Symbol?**

To represent the absolute value of a number (or a variable), we write a vertical bar on either side of the number. For example, the absolute value of 4 is written as |4|. Also, the absolute value of -4 is written as |-4|. As we discussed earlier, the absolute value results in a non-negative value all the time. Hence, |4|=|-4| =4. That is, it turns negative numbers also into positive numbers. The following figure represents the absolute value symbol.

**Important Notes**

The following summary points help in representing the absolute values.

- The absolute value of x is represented by either |x| or abs(x).
- The absolute value of any number always results in a non-negative value.
- We pronounce |x| as ‘mod x’ or ‘modulus of x.’

**Absolute Value Function**

The absolute value function is defined as f(x) = |x|, { |x| = +x for x > 0, and |x| = -x for x < 0} Using the definition of absolute value, we know that it always results in a non-negative number. Thus, the graph of f(x) = |x| looks as follows.

From the definition of absolute value function, the value of |x| depending on the sign of x. |x|= + x. We also know that √ {x^{2}} = + x. Therefore we have √{x^{2}} = | x |.

### Equations with one absolute value

#### A GENERAL NOTE: ABSOLUTE VALUE EQUATIONS

The absolute value of *x *is written as . It has the following properties:

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

An **absolute value equation** in the form has the following properties:

#### HOW TO: GIVEN AN ABSOLUTE VALUE EQUATION, SOLVE IT.

- Isolate the absolute value expression on one side of the equal sign.
- If , write and solve two equations: and .

Isolate the absolute value expression and then write two equations.

|3x−5|−4=6

|3x−5|=10

3x−5=10

3x=15

x=5

3x−5=−10

3x=−5

x=−5/3

There are two solutions: x=5, x=−5/3.

d. |−5x+10|=0

The equation is set equal to zero, so we have to write only one equation.

−5x+10=0

−5x=−10

x=2

There is one solution: x=2.

Solve the absolute value equation: |1−4x|+8=13.

**Solution**

x=−1, x=3/2

### Equations with two absolute values

Some of our absolute value equations could be of the form |u|=|v| where u and v are algebraic expressions. For example,

|x−3|=|2x+1|.

How would we solve them? If two algebraic expressions are equal in absolute value, then they are either equal to each other or negatives of each other. The property for absolute value equations says that for any algebraic expression, u, and a positive real number, a, if |u|=a, then u=a or u=−a.

This leads us to the following property for equations with two absolute values:

**EQUATIONS WITH TWO ABSOLUTE VALUES**

For any algebraic expressions, u and v, if |u|=|v|, then:

u=v or u=−v.

When we take the opposite of a quantity, we must be careful with the signs and to add parentheses where needed.

### Absolute value equations with no solutions

As we are solving absolute value equations it is important to be aware of special cases. An absolute value is defined as the distance from 0 on a number line, so it must be a positive number. When an absolute value expression is equal to a negative number, we say the equation has no solution, or DNE. Notice how this happens in the next two examples.

EXAMPLE

Solve for x. 7+|2x−5|=4

Result of absolute value is negative! The result of an absolute value must always be positive, so we say there is no solution to this equation, or DNE.

### Solve Inequalities Containing Absolute Values

Let’s apply what you know about solving equations that contain absolute values and what you know about inequalities to solve inequalities that contain absolute values. Let’s start with a simple inequality.

|x|≤4

This inequality is read, “the absolute value of xxis less than or equal to 4.” If you are asked to solve for x, you want to find out what values of x are 4 units or less away from 0 on a number line. You could start by thinking about the number line and what values of x would satisfy this equation.

4 and −4 are both four units away from 0, so they are solutions. 3 and −3 are also solutions because each of these values is less than 4 units away from 0. So are 1 and −1, 0.5 and −0.5, and so on—there are an infinite number of values for x that will satisfy this inequality.

The graph of this inequality will have two closed circles, at 4 and −4. The distance between these two values on the number line is colored in blue because all of these values satisfy the inequality.

The solution can be written this way:

Inequality: −4≤x≤4

Interval: [−4,4]

The situation is a little different when the inequality sign is “greater than” or “greater than or equal to.” Consider the simple inequality |x|>3. Again, you could think of the number line and what values of xx are greater than 3 units away from zero. This time, 3 and −3 are not included in the solution, so there are open circles on both of these values. 2 and −2 would not be solutions because they are not more than 3 units away from 0. But 5 and −5 would work, and so would all of the values extending to the left of −3 and to the right of 3. The graph would look like the one below.

The solution to this inequality can be written this way:

Inequality*:* x<−3x<−3 or x>3x>3.

Interval: (−∞,−3)∪(3,∞)

#### Writing Solutions to Absolute Value Inequalities

For any positive value of a and x*,* a single variable, or any algebraic expression:

Let’s look at a few more examples of inequalities containing absolute values.

EXAMPLE

Solve for x. |x+3|>4

**Solution**

Since this is a “greater than” inequality, the solution can be rewritten according to the “greater than” rule.

Check the solutions in the original equation to be sure they work. Check the end point of the first related equation, −7 and the end point of the second related equation, 1.

#### Identify cases of inequalities containing absolute values that have no solutions

As with equations, there may be instances in which there is no solution to an inequality.

### Solved Examples on Absolute Value

**Example 1: Mohan wants to find the values of the following. Shall we help her using the definition of absolute value? (I) |-13/5|, (II) – | -3|, (III) |2(-3) +4| .**

**Solution:**

We know that absolute value results in non-negative values all the time. Thus, we have the following solutions.

(I) | -13/15| = 13/15

(II). – |-3| = -(3) = -3

(III). |2(-3) +4| = |-6+4| = |-2|=2

Therefore(I) | -13/15| = 13/15, (II) – | -3 | = -3, (III) |2(-3) + 4| = 2

**Example 2: Ria is instructed by her teacher to solve the following absolute value equation using the definition of the absolute function. |x-2|=4. Can we try to help her?**

**Solution:**

The given equation is |x-2|=4. Using the definition of the absolute value function, when we remove the absolute value sign on one side of the equation. We then get __+__ sign on the other side. x-2= __+__ 4. This results in two equations, which we solve separately.

x – 2 = +4 x = +4 + 2 x = +6 | x – 2 = -4 x = -4 + 2 x = -2 |

Therefore, the solutions of the given equation are x = 6, x = -2.

**Example 3: Rohan wants to find the value of -|-x| when x>0. How can we help Rohan to find the absolute value?**

**Solution:**

Let’s help him using the definition of the absolute value function. It is given that x>0, then we have, -x<0. Now, byy the definition of the absolute value function we have: |-x|= -(-x) = x. Therefore, -|-x| = -x.

### FAQs on Absolute Value

**What is the absolute value of |-11|?**

The absolute value of |-11| is 11 because absolute value turns negative numbers into positive numbers, i.e., |-11|=11

#### What is Meant by Absolute Value?

The absolute value only gives the numeric value and does not show any sign. The absolute value of |5| is 5, and the absolute value of |-3| is 3.

#### How Do you Find the Absolute Value of a Negative Number?

The absolute value of a negative number is also a positive value. | -2| = 2. Irrespective of the sign of the numeric value, the absolute value is always positive.

#### What Is the Use of Absolute Value?

The absolute value is used to inform the numeric value of a quantity, irrespective of the sign of the quantity. Numerous quantities such as length, price, volume, do not signify any meaning for the sign and are written without any sign. Here the concept of absolute value is helpful to represent such quantities.

#### What is the Absolute Value of a Negative Integer?

The absolute value of a negative integer is also a positive value. | – 5| = 5. For example, the distance value is sometimes written as -5 meters, but the -ve sign only means the direction, and the distance is only 5 meters.

#### Can the Absolute Value be Negative?

The absolute value is always positive. Even for a positive or negative value within the modulus, the absolute value is always positive. |__+__X| = X.

#### Can Two Different Numbers have the Same Absolute Value?

The two numbers can also have the same absolute value. For example, the two numbers -7, or +7 have the same absolute value of 7. | -7| = |+7| = 7.

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