Absolute Value In An Equation

Absolute Value Function

An absolute value function is an of import function in algebra that consists of the variable in the absolute value confined. The full general form of the absolute value function is f(ten) = a |x - h| + one thousand and the most commonly used class of this function is f(10) = |x|, where a = 1 and h = thousand = 0. The range of this office f(x) = |x| is always non-negative and on expanding the absolute value office f(x) = |x|, nosotros can write information technology as x, if 10 ≥ 0 and -x, if ten < 0.

In this article, we will explore the definition, various properties, and formulas of the absolute value role. We will learn graphing accented value functions and determine the horizontal and vertical shifts in their graph. We shall solve various examples based related to the role for a better understanding of the concept.

1.	What is Accented Value Function?
two.	Absolute Value Office Definition
3.	Accented Value Office Graph
4.	Absolute Value Equation
5.	Graphing Absolute Value Functions
half-dozen.	FAQs on Absolute Value Function

What is Accented Value Function?

An absolute value part is a part in algebra where the variable is inside the accented value bars. This role is also known as the modulus function and the nigh normally used form of the absolute value function is f(x) = |x|, where 10 is a real number. More often than not, we tin represent the absolute value function as, f(10) = a |x - h| + 1000, where a represents how far the graph stretches vertically, h represents the horizontal shift and thousand represents the vertical shift from the graph of f(x) = |x|. If the value of 'a' is negative, the graph opens downwards and if it is positive, the graph opens upwards.

Absolute Value Office Definition

The absolute value function is defined equally an algebraic expression in absolute bar symbols. Such functions are commonly used to find distance between 2 points. Some of the examples of accented value functions are:

f(10) = |10|
thou(x) = |3x - vii|
f(x) = |-x + 9|

All the above given absolute value functions have non-negative values, that is, their range is all real numbers except negative numbers. All these functions alter their nature (increasing or decreasing) after a betoken. We can find those points by expressing the accented value function f(10) = a |10 - h| + grand as,

f(x) = a (10 - h) + 1000, if (x - h) ≥ 0 and

= – a (x - h) + k, if (x - h) < 0

Accented Value Function Graph

In this section, nosotros will understand how to plot the graph of the common form of the absolute value part f(10) = |x| whose formula can too be expressed as f(x) = x, if x ≥ 0 and -x, if x < 0. Permit us consider different points and determine the value of the function using the formula and plot them on a graph.

x	f(x) = \|x\|
-five	5
-4	iv
-3	3
-2	ii
-i	1
0	0
i	1
2	2
three	3
4	4
5	v

Absolute Value Function graph

Absolute Value Equation

Now that we accept understood the significant of the accented value role, now we volition sympathise the significant of the absolute value equation f(x) = a |ten - h| + k and how the values of a, h, thou affect the value of the function.

The value of 'a' determines how the graph of f(x) stretches vertically
The value of 'h' tells the horizontal shift
The value of 'k' tells the vertical shift

The vertex of the absolute value equation f(x) = a |10 - h| + k is given by (h, chiliad). We can also find the vertex of f(x) = a |x - h| + thousand using the formula (ten - h) = 0. On determining the value of x, we substitute the value into the equation to find the value of g.

Allow us consider an example and find the vertex of an absolute value equation.

Example 1: Consider the modulus function f(x) = |10|. Discover its vertex.

Solution: Compare the function f(10) = |x| with f(x) = a |x - h| + grand. We have a = 1, h = k = 0. And then, the vertex of the function is (h, k) = (0, 0).

Case 2: Observe the vertex f(ten) = |10 - 7| + 2.

Solution: On comparing f(x) = |x - 7| + ii with f(x) = a |10 - h| + k, nosotros take the vertex (h, k) = (7, 2).

We tin find it using the formula. So, nosotros have (x - 7) = 0

⇒ x = 7

Now, substitute x = 7 into the equation f(x) = |x - seven| + ii, we take

f(ten) = |7 - 7| + 2

= 0 + two

= 2

And then, the vertex of absolute value equation f(x) = |x - 7| + ii using the formula is (seven, 2).

Graphing Absolute Value Functions

In this section, we will learn graphing absolute value functions of the form f(x) = a |x - h| + m. The graph of an absolute value function is ever either 'V-shaped or inverted 'V-shaped depending upon the value of 'a' and the (h, m) gives the vertex of the graph. Let us plot the graph of two accented value functions below.

f(ten) = two |x + 2| + 1 and g(ten) = -two |x - 2| + iii

On comparing the two absolute value functions with the full general class, a is positive in f(x), so it will open upwardly and its vertex is (-2, 1). For g(x), the value of a = -2 which is negative, so the graph will open downward and its vertex is (2, three). The image below shows the graph of the absolute value functions f(10) and thou(10).

graphing absolute value functions

Important Notes on Absolute Value Function

The general form of the accented value function is f(x) = a |x - h| + one thousand, where (h, m) is the vertex of the graph.
An accented value function is a office in algebra where the variable is inside the absolute value confined.
The graph of an accented value part is e'er either '5-shaped or inverted 'Five-shaped depending upon the value of 'a'.

☛ Related Articles:

Abiding Part
Inverse of a Function
Graphing Functions

Absolute Value Office Examples

Example 1: Detect the vertex of the absolute value function f(x) = -2 |x - 1| - 3

Solution: On comparing the function f(x) = -two |x - ane| - 3 with f(x) = a |ten - h| + m, the vertex of the role is (h, k) = (1, -three)

Reply: Vertex = (i, -iii)
Example 2: Plot the graph of absolute value function f(10) = - |ten + 2| - 3

Solution: As we can see, the value of a is -1 in f(ten) = - |x + 2| - iii which is negative. SO, the graph opens downwards and hence will be inverted V-shaped. The vertex of the graph is (h, grand) = (-ii, -iii).

So, the graph of the given absolute value function f(x) = - |x + ii| - 3 is given by,
Example 3: Notice the derivative of the accented value function f(x) = |x|.

Solution: We tin can write f(ten) = |10| every bit,

f(x) = 10, if x ≥ 0 and

f(x) = -10, if x < 0

At present, nosotros know that the derivative of x is 1 and the derivative of -10 is -one. So, the derivative of f(ten) = |ten| is given by,

d(|ten|)/dx = 1, if x > 0 and

= -one, if x < 0

Answer: d(|x|)/dx = i, if x > 0 and -1, if x < 0.

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FAQs on Absolute Value Function

What is Absolute Value Part?

An absolute value function is an of import function in algebra that consists of the variable in the absolute value confined. The general form of the absolute value function is f(ten) = a |ten - h| + k, where (h, one thousand) is the vertex of the part.

What is an Example of Absolute Value Role?

Some of the examples of absolute value functions are:

f(10) = |x|
g(x) = 2 |3x - 5| + 5
f(x) = |-10 - ix|
f(x) = 3 |x|

How To Find the Vertex of an Absolute Value Office?

The full general form of the absolute value role is f(x) = a |x - h| + 1000, where (h, thousand) is the vertex of the function. Then, to notice the vertex of the part, nosotros compare the two equations and determine the values of h and k.

What Does the Value of k Do to the Absolute Value Function?

The value of 'yard' in f(10) = a |10 - h| + k tells us the vertical shift from the graph of f(x) = |x|. The graph moves upwards if k > 0 and moves downwards if 1000 < 0.

Why is An Absolute Value Part Not Differentiable?

An absolute value function f(x) = a |ten - h| + k is not differentiable at the vertex (h, 1000) because the left-paw limit and the correct-hand limit of the function are not equal at the vertex.

Is an Absolute Value Function Fifty-fifty or Odd?

The absolute value function f(x) = |x| is an even office considering f(10) = |x| = |-x| = f(-x) for all values of x.

How to Write an Absolute Value Function as a Piecewise Function?

We can write the absolute value function f(x) = |x| every bit a piecewise function as, f(10) = x, if ten ≥ 0 and -x, if x < 0.