Graphing absolute value functions

Graphing Absolute Value Functions

Absolute value functions are defined as functions that measure the distance of a number from zero on the number line. The absolute value of a number is always non-negative. The notation for absolute value is |x|, where x is the number. The absolute value function can be expressed mathematically as:

|x| = { x, if x ≥ 0
-x, if x < 0 }

This means that if the value of x is positive or zero, the absolute value function returns x. If x is negative, it returns the opposite of x (which is positive).

Basic Shape of Absolute Value Functions

The graph of the absolute value function |x| has a characteristic V shape. The vertex of the V is at the origin (0, 0). As we move away from the origin in both directions (positive and negative), the value of the function increases. The function is symmetric about the y-axis, meaning that it looks the same on both sides of the y-axis.

Graphing the Basic Absolute Value Function

To graph the function y = |x|, follow these steps:

1. Create a table of values for x and y.
2. Plot the points on a Cartesian plane.
3. Connect the points to form the V shape.

Here is an example of creating a table of values for the function y = |x|:

x | y
---|---
-3 | 3
-2 | 2
-1 | 1
 0 | 0
 1 | 1
 2 | 2
 3 | 3

Now, plot the points (-3, 3), (-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2), and (3, 3) on the Cartesian plane. Connect these points to form the V shape.

Transformations of Absolute Value Functions

Absolute value functions can be transformed by changing their equations. The general form of a transformed absolute value function is:

y = a |x - h| + k

In this equation:

• a affects the vertical stretch or compression and the direction of the graph.
• h shifts the graph horizontally.
• k shifts the graph vertically.

Vertical Stretch and Compression

The value of a determines if the graph is stretched or compressed. If |a| > 1, the graph is stretched. If 0 < |a| < 1, the graph is compressed. If a is negative, the graph opens downward.

Horizontal Shift

The value of h shifts the graph left or right. If h is positive, the graph shifts to the right. If h is negative, the graph shifts to the left.

Vertical Shift

The value of k shifts the graph up or down. If k is positive, the graph shifts up. If k is negative, the graph shifts down.

Example of Transforming an Absolute Value Function

Consider the function y = 2 |x - 1| + 3. We will graph this function by identifying the transformations:

• The value of a is 2, which means the graph will be stretched vertically by a factor of 2.
• The value of h is 1, which means the graph shifts to the right by 1 unit.
• The value of k is 3, which means the graph shifts up by 3 units.

To graph this function, we can follow these steps:

1. Identify the vertex: The vertex is at (h, k) = (1, 3).
2. Create a table of values around the vertex:

x | y
---|---
0 | 5
1 | 3
2 | 3
3 | 5

Now, plot the points (0, 5), (1, 3), (2, 3), and (3, 5) on the Cartesian plane. Connect these points to form the transformed V shape.

Graphing Absolute Value Functions with Negatives

When the absolute value function includes a negative sign, such as y = -|x|, the graph will open downward. The vertex remains at (0, 0), but the graph will reflect across the x-axis.

For example, to graph y = -|x|:

1. Create a table of values:

x | y
---|---
-3 | -3
-2 | -2
-1 | -1
 0 | 0
 1 | -1
 2 | -2
 3 | -3

Now, plot the points (-3, -3), (-2, -2), (-1, -1), (0, 0), (1, -1), (2, -2), and (3, -3) on the Cartesian plane. Connect these points to form the inverted V shape.

Summary

• The graph of the absolute value function |x| has a V shape with the vertex at (0, 0).
• Transformations can stretch, compress, or shift the graph.
• The general form of a transformed absolute value function is y = a |x - h| + k.
• Negative absolute value functions open downward.

Check your understanding

1. What is the vertex of the function y = 3|x + 2| - 4?
2. How does the value of a affect the graph of an absolute value function?
3. What is the effect of a negative sign in front of the absolute value function?
4. Graph the function y = |x - 1| + 2 and describe its transformations.