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# Graphing Absolute Value Functions

Graphing of a function gives a pictorial representation of the function. The absolute value always gives a positive value. The absolute function graph will be of V-shape and its position can be predicted on the basis of certain rules about graphing the absolute value functions. The graph can be shifted, stretched, shrank and reflected on the basis of its properties.

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## Definition

An absolute value function is in the form y = |f(x)| + c. The graph of an absolute value function will be of V-shape. If the graph is of the functions y = |x| or x = |y| then the V-shape will start from the origin. But if the function is of the form of y = |x| + c, y = |x + c|, x = |y| + c and x = |y + c| there will be shifting in the V-shaped graph from the origin.

The graph of y = |x| will be as given below:

The graph for x = |y| is as given below:

## Absolute Value Graphs Shifts

There can vertical and horizontal shifts in a graph. Given are the rules to know the shift in the graph of an absolute value function.

1. y = |x| + c: It will shift the graph c points upwards.

In the given example, the function is y = |x| + 2 which shifts the graph 2 points upward.

2. y = |x| - c: It will shift the graph c points downwards.

In the given example, the function is y = |x| - 2 which shifts the graph 2 points downward.

3. y = |x + c|: It will shift the graph c points leftwards.

In the given example, the function is y = |x + 2| which shifts the graph 2 points leftward.

4. y = |x - c|: It will shift the graph c points rightwards.

In the given example, the function is y = |x - 2| which shifts the graph 2 points rightward.

If the graph of an absolute value function is multiplied by -1, then it will be reflected over the axis.

## Stretching and Shrinking the Graph of Absolute Value Functions

Let the graph be of the absolute value function y = |x|. The graph will be stretched or compressed based on the given rules.

1. y = c|x|: The graph will stretch c times.

2.
y = $\frac{1}{c}$|x|: The graph will be compressed c times.

3. y = $|\frac{1}{c}x|$: The graph will be stretched c times.

4. y = |cx|: The graph will be compressed c times.

## Examples

Example 1: Predict the behavior of the graph of the function, y = |3x| - 5.

Solution: The graph will be V-shaped symmetric to the X-axis.

It will be compressed by 3.

It will shift 5 points below the origin.

The graph can be plotted as here.

Example 2: Draw the graph of x = |5y - 2|.

Solution: Given function is x = |5y - 2|

As we saw above, a graph of the function like y = |x - c| will shift the graph c points rightwards. Similarly graph of the function like x =
|y - c|
will shift the graph c points upwards(on x = 0).

The graph of the given function is as below:

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