An absolute value function is a mathematical function that describes the distance between a number and zero on the number line. It is also known as a modulus function, and its symbol is represented by two vertical lines enclosing a number, such as |x|.

In simpler terms, the absolute value function returns the positive value of a number, regardless of its sign. For instance, the absolute value of -5 is 5, and the absolute value of 5 is also 5.

To plot the graph of an absolute value function, select multiple values of x and determine the corresponding ordered pairs.

x | y=|x| |

-2 | 2 |

-1 | 1 |

0 | 0 |

1 | 1 |

2 | 2 |

To graph it, plot the identified ordered pairs on a coordinate plane, and then connect them to form the graph.

Note that the graph of the this function has a V-shape.

- The origin (0,0) is the vertex of the absolute value function graph.
- The axis of symmetry is the y-axis (x=0), dividing the graph into two identical parts.
- The domain of the function is all real numbers.
- The range of the function is all real numbers greater than or equal to zero (y≥0).
- The x-intercept and y-intercept of the graph are both located at (0,0).

**Horizontal Shift**

To perform a horizontal translation of the absolute value function f(x) = |x|, the function g(x) = f(x-h) can be used.

If h>0, the graph of f(x) is shifted to the right by h units to obtain the graph of g(x).

If h<0, the graph of f(x) is shifted to the left by |h| units to obtain the graph of g(x).

**Vertical Shift**

To perform a vertical translation of this function f(x) = |x|, the function g(x) = f(x) + k can be used.

If k>0, the graph of g(x) is shifted upwards by k units.

If k<0, the graph of g(x) is shifted downwards by |k| units.

**Stretch and Compression**

The stretching or compressing of y = |x| is determined by the function y = a|x|, where a is a constant.

- If a > 0, the graph is stretched vertically, and it opens upwards.
- If a < 0, the graph is compressed vertically, and it opens downwards.

For absolute value equations of the form y = a|x|, where a is a constant:

- If 0<a<1, the graph is vertically compressed.
- If a>1, the graph is vertically stretched.
- If a<0, the graph is reflected about the x-axis and opens downwards instead of upwards as usual.

The general form of an absolute value function equation is y = a| x−h |+k. Additionally:

- The vertex of the graph is located at (h,k).
- The domain of the function is all real numbers and the range is y≥k if a>0.
- The domain of the function is all real numbers and the range is y≤k if a<0.
- The axis of symmetry is the vertical line x=h.
- The graph opens upwards if a>0 and downwards if a<0.

To obtain the graph of y = a| x−h |+k, the graph of y = | x | is translated h units horizontally and k units vertically.

The graph of y = a| x | is wider than the graph of y = | x | if | a |<1 and narrower if |a|>1.

The absolute value function is commonly used in mathematical equations, including algebra, calculus, and geometry. Here are some important aspects of absolute value functions:

**Graphing Absolute Value Functions**

When graphing absolute value functions, the resulting graph appears as a V-shape, also known as a “mountain” or “valley.” The vertex of the V-shape is the point at which the graph changes direction. It is also the lowest or highest point on the graph, depending on the sign of the coefficient.

The equation of function is f(x) = |x – h| + k, where h and k represent the horizontal and vertical shifts, respectively. The coefficient of the function determines the shape and direction of the graph.

**Solving Absolute Equations**

Solving absolute value equations involves finding the values of a variable that satisfy the equation. To solve an absolute value equation, we must isolate the absolute value expression and consider both the positive and negative solutions.

For instance, to solve |2x – 3| = 5, we can rewrite it as 2x – 3 = 5 or 2x – 3 = -5. Solving each equation yields x = 4 or x = -1, respectively.

**Applications of Absolute Value Functions**

It has several practical applications, including in physics, engineering, and finance. For instance, in physics, the velocity of an object can be modelled using an absolute value function to represent the speed at which the object is moving.

In finance, this function is used to calculate the absolute return of an investment, which measures the percentage change in the value of the investment regardless of whether it is positive or negative.

**Properties of Absolute Value Functions**

These functions have several essential properties that make them useful in mathematical applications. These include the following:

- The absolute value of a positive number is the number itself, while the absolute value of a negative number is its opposite.
- The absolute value function is continuous for all real numbers
- The absolute value function is non-negative, meaning it can never be less than zero

**Absolute Value Properties**

The absolute value has several properties, including:

**Non-negativity: **

The absolute value of any real number a is always greater than or equal to zero: |a| ≥ 0.

**Positive-definiteness: **

The absolute value of a is equal to zero if and only if a is zero: |a| = 0 if and only if a = 0.

**Multiplicativeness: **

The absolute value of the product of two real numbers a and b is equal to the product of their absolute values: |ab| = |a||b|.

**Subadditivity: **

The absolute value of the sum of two real numbers a and b is less than or equal to the sum of their absolute values: |a + b| ≤ |a| + |b|.

Other important properties of the absolute value include:

**Idempotence: **

The absolute value of the absolute value of a real number a is equal to the absolute value of a: ||a|| = |a|.

**Symmetry: **

The absolute value of the negation of a real number a is equal to the absolute value of a: |-a| = |a|.

**Identity of indiscernibles: **

The absolute value of the difference of two real numbers a and b is equal to zero if and only if a is equal to b: |a – b| = 0 if and only if a = b.

**Triangle inequality: **

The absolute value of the difference of two real numbers a and b is less than or equal to the sum of the absolute values of their differences with a third real number c: |a – b| ≤ |a – c| + |c – b|.

**Preservation of division: **

The absolute value of the quotient of two real numbers a and b is equal to the quotient of their absolute values, as long as b is not zero: |a/b| = |a|/|b| if b ≠ 0.

Inequalities involving absolute values can be solved using the following useful properties:

|a| ≤ b if and only if -b ≤ a ≤ b.

|a| ≥ b if and only if a ≤ -b or a ≥ b.

For example, the inequality |x – 3| ≤ 9 can be solved as follows:

| x – 3 | ≤ 9

-9 < x – 3 < 9 (using property 10)

-6 < x < 12 (adding 3 to all sides)

These properties are useful in many different areas of mathematics and are often used in solving problems involving absolute values.

**Example 1**

Find the vertex of the absolute value function g(x) = 3 |x + 2| – 1

**Solution:**

let’s consider the function g(x) = 3 |x + 2| – 1.

Here, a = 3, h = -2, and k = -1. So, the vertex is located at the point (h,k) = (-2,-1).

Therefore, the vertex of the absolute value function g(x) = 3 |x + 2| – 1 is (-2,-1).

**Example 2**

Find the derivative of the absolute value function f(x) = |x|.

**Solution**

The derivative of the function f(x) = |x| is not defined at x = 0 because the function is not differentiable at that point. However, we can find the derivative for x > 0 and x < 0 separately.

For x > 0, the absolute value function is simply f(x) = x, and its derivative is:

f'(x) = d/dx (x) = 1

For x < 0, the absolute value function is f(x) = -x, and its derivative is:

f'(x) = d/dx (-x) = -1

Therefore, the derivative of the absolute value function f(x) = |x| is:

f'(x) = 1 for x > 0 f'(x) = -1 for x < 0

Note that at x = 0, the derivative does not exist.

**Example 3**

What is the value of 3 | 4x + 2 | if x = 3

**Solution**

To solve this problem, we need to substitute x = 3 into the expression 3 | 4x + 2 | and simplify.

Step 1: Substitute x = 3 into the expression

3 | 4x + 2 | = 3 | 4(3) + 2 | = 3 | 12 + 2 | = 3 | 14 |

Step 2: Simplify the absolute value

The absolute value of 14 is simply 14, since 14 is already a positive number.

3 | 14 | = 3(14) = 42

Therefore, the value of 3 | 4x + 2 | when x = 3 is 42.

**Example 4**

Solve 2 | x + 3 | = 10

**Solution**

To solve this problem, we need to isolate the absolute value expression and then solve for the value of x.

Step 1: Isolate the absolute value expression

We can start by dividing both sides of the equation by 2:

2 | x + 3 | = 10 | x + 3 | = 5

Step 2: Solve for x

Now we have an equation with an absolute value on one side and a number on the other. To solve for x, we need to consider two cases:

Case 1: x + 3 is positive

If x + 3 is positive, then we can simply remove the absolute value bars:

| x + 3 | = x + 3

Substituting this into our equation, we get:

x + 3 = 5

Solving for x, we get:

x = 2

Case 2: x + 3 is negative

If x + 3 is negative, then the absolute value of x + 3 is simply the negative of x + 3:

| x + 3 | = -(x + 3)

Substituting this into our equation, we get:

-(x + 3) = 5

Solving for x, we get:

x = -8

Step 3: Check the solutions

To check our solutions, we need to substitute them back into the original equation and verify that they make the equation true:

When x = 2:

2 | x + 3 | = 10 2 | 2 + 3 | = 10 2 | 5 | = 10 10 = 10

When x = -8:

2 | x + 3 | = 10 2 | -8 + 3 | = 10 2 | -5 | = 10 10 = 10

Both solutions check out, so our final answer is:

x = 2 or x = -8

In conclusion, the absolute value function is a vital mathematical concept that is widely used in various fields. It provides a useful tool for graphing, solving equations, and modelling real-world scenarios. Understanding its properties and applications can help individuals appreciate its importance in mathematics and beyond.