# Absolute value graph and function review (article) | Khan Academy (2023)

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.

## Want to join the conversation?

• is there any easier steps to explain this type of lesson

• Maybe I can better explain

when you have an absolute value function you want to look at what are in the places of a, h and k. a|x-h|+k Specifically you want to look at h and k first. Normally the tip of the V shape is at (0,0) this changes depending on h and k. specifically it moves the tip to (h,k) so if you have |x+5|-7 then the tip of the V shape goes to (-5,-7). if you wonder why it is -5 even though we are adding 5, you just need to look at the original a|x-h|+k if we had -5 then it would be just like that, but since it is +5, we have to look at it as - -5, minus negative 5. so if it helps, the x coordinate is kinda backwards.

After the V tip you then look at a. treat it like a linear equation where a is the slope. so if a was -3 that's down 3 right 1 using rise over run. then, since it's an absolute value function you need to know that the same line goesalong the left to make that V shape, so -5 would mean on the left down 3 and left 1.

if you ever have something like a|bx-h|+k where there is a number in front of the x you need to get rid of it if you are not aware of factoring this is what it would look like a|b||x - h/b|+k where a|b| becomes the new "a" and h/b becomes the new h, then you would solve it normally. The point being you always want x by itself for this. Also, keep in mind that even if inside the absolute value bars if b was negative, outside it becomes positive.

Let me know if that didn't help, or if there is a specific function you are struggling with, or maybe would even like some to try out.

• In example problem 1, why isn’t the graph shifted 1 unit to the left instead of to the right?

• It is shifted to the right because `x-1` would make it `0` when `x=1` because
`x=1`
`1-1=0`
So, we always want the absolute value part of the equation to be equal to 0 when we use x as the horizontal shifting.

While, the vertical part goes up with + not down because when,
`y=a∣x−h∣+k`
`y-k=a|x-h|`
So basically we transpose it to make it easier to distinguish.

• He is right next to you

• How would we utilize this in real life? For what careers?

• mathematician

• If someone needs:
Horizontal shift : y = f(x+b)
Vertical shift: y = f(x) +d
Reflection about the X-axis : y = -f(x)
Reflection about the Y-axis : y = f(-x)
Stretch/Compress in the X direction: y = f(a * x)
Stretch/Compress in the Y direction: y=f(x) * a

• So is h is positive that means that it is actually negative? Is that why if its x + 3 on the graph you go to negative 3?

• No, if it is positive it means I move in the negative direction, but if h is negative I move in the positive direction, it does not change the sign of h. The idea is that what value of x would make the inside of the absolute value (or other function) 0, so if you have x + 3, it would require x = -3 to be 0, thus causing a shift in the negative direction. The other idea is that since the formula has | x - h | + k where (h,k) is the vertex, then using x+ 3 would actually be x - (- 3) so -3 would cause a shift to the left.

• I am confused on how you know if the vertex is a minimum or maximum point.

(1 vote)

• The general form of the absolute value function is:
f(x) = a|x-h|+k
When "a" is negative, the V-shape graph opens downward and the vertex is the maximum.
When "a" is positive, the V-shape graph opens upward and the vertex is a minimum.
Hope this helps.

• i dont know what made the diffrence to make it go up or down

(1 vote)

• y=a|x-h|+k
here 'a' is the slope of the line. If slope (which is 'a') is positive the line will go upward. As we are dealing with the absolute of x (|x|) it won't affect the value of 'a'.

(1 vote)

• What if there's something in front of the x? In my homework, there's a problem that says a(x) = |5x|. How would I graph that?

• You would simplify the expression inside the `| |` and take the absolute value of the result. For example
|5*0| = |0| = 0
|5*1| = |5| = 5
|5*-1| = |-5| = 5
|5*2| = |10| = 10
|5*-2| = |-10| = 10

• I have a table where the coordinates are in a table, the coordinates are
(4,7) (5,6) (6,5) (7,4) (8,5)
it is wanting me to find the domain which I found, but I don't know how to find the range, intercepts, vertex, or the max or min.
How do I find the range, intercepts, vertex, and the max or min?

• The range refers to the possible y values. Based on the table you gave us, it looks like there is a minimum y value. It looks like the y values start at 7, go down, and then start going up again. The lowest y value is going to be the turning point of the variable. The turning point is always going to be the minimum or the maximum. The turning point is your vertex.

Since there appears to be a lowest y value, the graph probably doesn't have a highest y value. It depends a little on the question.

The vertex is where the graph turns. It should happen at the lowest y value (or the highest, if that makes more sense for the problem. If the graph goes up forever, then it should be the lowest y value). The vertex is also where the min, or max of the function is... but remember, whatever the vertex is (max or min), the other one (min or max) is probably infinity.

For intercepts... based on the table, does it look like it ever crosses the x axis? Use the slope of the line (before it turns, or after it turns), to figure out where the line goes beyond the points that it gave you. For the Y-intercept, what is the Y value when X is 0?

## 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.

How do you find the answer to an absolute value using a graph? ›

To solve absolute value equations, find x values that make the expression inside the absolute value positive or negative the constant. To graph absolute value functions, plot two lines for the positive and negative cases that meet at the expression's zero. The graph is v-shaped.

What is the equation for the absolute value function Khan Academy? ›

Absolute value graphs review. The general form of an absolute value function is f(x)=a|x-h|+k.

Do all absolute value problems 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.

What grade level is absolute value equations? ›

IXL | Solve absolute value equations | 8th grade math.

Why do most absolute value equations have two answers? ›

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

When can you not solve absolute value? ›

An absolute value equation has no solution if the absolute value expression equals a negative number since an absolute value can never be negative.

How do you know how many solutions an absolute value equation has? ›

First, determine the term at the right hand side of the equation. If the right hand side contains zero, then it has one solution. If it contains any other positive value, then the number of solutions is two.

What is an example of an absolute value function? ›

Some of the examples of absolute value functions are: f(x) = |x| g(x) = |3x - 7| f(x) = |-x + 9|

### What is an example of an absolute value function equation? ›

Absolute Value Function
x|x| = y(x,y)
-2|-2| = 2(-2,2)
-1|-1| = 1(-1,1)
0|0| = 0(0,0)
1|1| = 1(1,1)
1 more row
Sep 20, 2021

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

Can an absolute value be negative? ›

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 is an example of absolute value in Algebra 2? ›

Every real number is either positive or negative. The two numerals +7 and -7 have different signs but the numerical part of each is 7. This part of the numeral designates the absolute value of the number. The absolute value of a is denoted by | a | (a vertical bar on each side of the quantity).

Can two different numbers have the same absolute value? ›

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

What is the 7th grade definition for absolute value? ›

Absolute value describes the distance from zero that a number is on the number line, without considering direction. The absolute value of a number is never negative.

What age do you learn absolute value? ›

Absolute Value - Chapter Summary

The video lessons in this Absolute Value chapter are designed to help your 6th, 7th or 8th graders understand how to work with and graph absolute value equations.

What is absolute value grade 11 math? ›

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.

Why is the absolute value of a negative number always positive? ›

The absolute value of a number or integer is the actual distance of the integer from zero, in a number line. Therefore, the absolute value is always a positive value and not a negative number.

What makes an absolute value equation all real numbers? ›

If the absolute value is greater than or greater than or equal to a negative number, the solution is all real numbers. The absolute value of something will always be greater than a negative number.

### 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.

Are absolute value functions always even? ›

Absolute value function

Since a real number and its opposite have the same absolute value, it is an even function, and is hence not invertible. The real absolute value function is a piecewise linear, convex function.

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.

How do you solve absolute value without variables? ›

To solve an equation containing absolute value, isolate the absolute value on one side of the equation. Then set its contents equal to both the positive and negative value of the number on the other side of the equation and solve both equations.

What is an example of an absolute value with no solution? ›

For example, |x|=−1 has no solution. The absolute value of a number is its distance away from zero. That number will always be positive, as you cannot be negative two feet away from something. So any absolute value equation set equal to a negative number is no solution, regardless of what that number is.

What is the use of absolute value function in real life? ›

A geophysicist uses absolute value to look at the total amount of energy used. In an energy wave, there are both negative and positive directions of movement. Another example is when scuba divers discuss their location in regards to sea level. “50 feet below sea level” doesn't have to be represented as -50 feet.

Why is absolute value necessary? ›

We often use absolute values when talking about distance because there is no such thing as negative distance. For example, just because the store you're going to is a block behind you doesn't mean it's negative one blocks away.

Is absolute value always positive yes or no? ›

The absolute value of a number is always positive or zero. If the original number is negative, its absolute value is that number without the negative sign.

How do you convert a function to absolute value? ›

The absolute value function f(x) = a | x - h | + k translates the absolute value graph up or down.

How do you know how many solutions an absolute value has? ›

First, determine the term at the right hand side of the equation. If the right hand side contains zero, then it has one solution. If it contains any other positive value, then the number of solutions is two.

### 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 are the 3 steps to solving absolute value equations? ›

To solve an equation containing absolute value, isolate the absolute value on one side of the equation. Then set its contents equal to both the positive and negative value of the number on the other side of the equation and solve both equations.

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 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 a number Grade 7? ›

The absolute value of a number is its distance from zero on the number line.

What is the absolute value of -(- 11? ›

The distance for −11 will also equal 11 (because |−11 − 0| = |−11| = 11), and the absolute value of −11 is 11. Thus, the absolute value of any real number is equal to the absolute value of its distance from 0 on the number line.

Why absolute value equations can have no solution one solution or two solutions? ›

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.

Can two different numbers have the same absolute value explain? ›

Two numbers are opposites if they have the same absolute value but different signs. Opposites are the same distance from 0 on a number line, and they are on opposite sides of 0. The opposite of 0 is 0.

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