Writing An Absolute Value Equation

Absolute Value Office

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

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

1.	What is Absolute Value Role?
2.	Accented Value Part Definition
3.	Absolute Value Function Graph
4.	Absolute Value Equation
5.	Graphing Absolute Value Functions
vi.	FAQs on Accented Value Role

What is Accented Value Function?

An absolute value function is a function in algebra where the variable is inside the accented value bars. This role is also known as the modulus function and the near commonly used grade of the accented value function is f(10) = |x|, where ten is a real number. Generally, nosotros can represent the absolute value part as, f(x) = a |10 - h| + k, where a represents how far the graph stretches vertically, h represents the horizontal shift and k represents the vertical shift from the graph of f(x) = |x|. If the value of 'a' is negative, the graph opens down and if it is positive, the graph opens upwards.

Absolute Value Office Definition

The absolute value office is defined as an algebraic expression in absolute bar symbols. Such functions are usually used to find distance between two points. Some of the examples of absolute value functions are:

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

All the above given absolute value functions take 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 point. We can find those points by expressing the absolute value part f(x) = a |x - h| + k as,

f(ten) = a (x - h) + k, if (ten - h) ≥ 0 and

= – a (x - h) + k, if (x - h) < 0

Absolute Value Office Graph

In this section, we will understand how to plot the graph of the common form of the absolute value function f(x) = |ten| whose formula tin can as well exist expressed every bit f(x) = ten, if x ≥ 0 and -x, if ten < 0. Let united states of america consider dissimilar points and make up one's mind the value of the part using the formula and plot them on a graph.

x	f(x) = \|x\|
-5	5
-iv	4
-3	3
-2	2
-1	1
0	0
1	1
ii	2
iii	3
iv	4
5	5

Absolute Value Function graph

Absolute Value Equation

Now that we have understood the meaning of the accented value function, now we will sympathize the meaning of the accented value equation f(ten) = a |x - h| + yard and how the values of a, h, one thousand bear upon 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(10) = a |10 - h| + k is given past (h, k). We can also notice the vertex of f(x) = a |x - h| + k using the formula (ten - h) = 0. On determining the value of x, we substitute the value into the equation to observe the value of yard.

Permit us consider an example and notice the vertex of an absolute value equation.

Example 1: Consider the modulus office f(x) = |x|. Observe its vertex.

Solution: Compare the function f(x) = |ten| with f(ten) = a |x - h| + 1000. We have a = 1, h = grand = 0. So, the vertex of the function is (h, k) = (0, 0).

Instance two: Notice the vertex f(x) = |10 - 7| + two.

Solution: On comparison f(x) = |x - 7| + ii with f(ten) = a |x - h| + grand, we have the vertex (h, 1000) = (7, 2).

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

⇒ x = 7

Now, substitute x = 7 into the equation f(x) = |10 - 7| + 2, we take

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

= 0 + 2

= 2

So, the vertex of absolute value equation f(x) = |10 - vii| + 2 using the formula is (seven, 2).

Graphing Absolute Value Functions

In this section, nosotros will learn graphing absolute value functions of the form f(x) = a |10 - h| + k. The graph of an absolute value part is always either 'V-shaped or inverted 'Five-shaped depending upon the value of 'a' and the (h, k) gives the vertex of the graph. Allow us plot the graph of two accented value functions below.

f(ten) = 2 |10 + two| + 1 and k(ten) = -2 |x - 2| + 3

On comparison the two absolute value functions with the full general course, a is positive in f(x), and so it will open upwardly and its vertex is (-2, ane). For m(ten), the value of a = -2 which is negative, so the graph will open downwards and its vertex is (2, 3). The image below shows the graph of the accented value functions f(10) and g(ten).

graphing absolute value functions

Important Notes on Absolute Value Function

The general form of the accented value function is f(ten) = a |10 - h| + k, where (h, g) is the vertex of the graph.
An absolute value function is a part in algebra where the variable is inside the absolute value bars.
The graph of an absolute value function is ever either 'V-shaped or inverted 'Five-shaped depending upon the value of 'a'.

☛ Related Articles:

Constant Function
Inverse of a Role
Graphing Functions

Accented Value Function Examples

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

Solution: On comparison the function f(ten) = -ii |x - 1| - 3 with f(x) = a |10 - h| + 1000, the vertex of the function is (h, k) = (one, -three)

Answer: Vertex = (one, -3)
Case two: Plot the graph of absolute value office f(x) = - |x + 2| - 3

Solution: As we tin encounter, the value of a is -1 in f(x) = - |x + 2| - 3 which is negative. SO, the graph opens downwardly and hence will exist inverted V-shaped. The vertex of the graph is (h, 1000) = (-2, -3).

So, the graph of the given absolute value office f(x) = - |10 + 2| - 3 is given past,
Example 3: Detect the derivative of the absolute value function f(x) = |x|.

Solution: We tin write f(x) = |x| as,

f(x) = x, if x ≥ 0 and

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

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

d(|x|)/dx = 1, if 10 > 0 and

= -1, if x < 0

Answer: d(|x|)/dx = 1, if 10 > 0 and -1, if x < 0.

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Absolute Value Function Questions

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FAQs on Absolute Value Function

What is Absolute Value Function?

An absolute value function is an important function in algebra that consists of the variable in the absolute value bars. The general form of the absolute value function is f(x) = a |10 - h| + k, where (h, k) is the vertex of the function.

What is an Case of Absolute Value Function?

Some of the examples of absolute value functions are:

f(x) = |x|
g(ten) = 2 |3x - 5| + 5
f(x) = |-ten - 9|
f(x) = 3 |x|

How To Find the Vertex of an Absolute Value Office?

The general form of the absolute value function is f(ten) = a |x - h| + 1000, where (h, yard) is the vertex of the part. So, to find the vertex of the function, we compare the two equations and make up one's mind the values of h and grand.

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

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

Why is An Absolute Value Office Not Differentiable?

An absolute value function f(x) = a |x - h| + k is not differentiable at the vertex (h, g) because the left-mitt limit and the right-manus limit of the function are not equal at the vertex.

Is an Absolute Value Function Even or Odd?

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

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

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