Question

Use a graphing utility. Graph \(y=|x| .\) Then on the same screen graph \(y=2|x|,\) followed by \(y=4|x|,\) followed by \(y=\frac{1}{2}|x|\) What pattern do you observe? Can you predict the graph of \(y=\frac{1}{4}|x| ?\) Of \(y=5|x| ?\)

Short answer

As the coefficient of \(\|x\| \) increases, the graph becomes steeper. \(y=\frac{1}{4}|x| \) will be less steep, and \(y=5|x| \) will be steeper.

Step-by-step solution

Graph the base function

Graph the function \(y = |x| \) on the graphing utility. This will be a V-shaped graph that opens upwards and passes through the origin (0,0).

Graph the second function

Next, graph the function \(y = 2|x| \) on the same screen. Notice that this graph also has a V-shape and passes through the origin, but it is steeper than the graph of \(y = |x| \).

Graph the third function

Now, graph the function \(y = 4|x| \). Again, it is V-shaped and passes through the origin, but it is even steeper than both \(y = |x|\) and \(y = 2|x| \).

Graph the fourth function

Graph the function \(y = \frac{1}{2}|x| \). This graph will be V-shaped, passing through the origin, but it will be less steep than the graph of \(y = |x| \).

Observe the pattern

Observe the pattern in the steepness of the graphs: as the coefficient of \(|x| \) increases, the graph becomes steeper. As the coefficient decreases, the graph becomes less steep.

Predict the graph of y=\frac{1}{4}|x|

Based on the pattern, the graph of \(y = \frac{1}{4}|x| \) will be V-shaped, pass through the origin, and will be even less steep than \(y = \frac{1}{2}|x| \).

Predict the graph of y=5|x|

For \(y = 5|x| \), predict that the graph will be V-shaped, pass through the origin, and be steeper than \(y = 4|x|\).

Key concepts

absolute value function

An absolute value function is represented as \(|x|\). It outputs the distance of a number from zero on the number line, regardless of direction. So, for a given input x, the absolute value function will return x if x is positive or zero, and -x if x is negative.
Graphically, the function \(y = |x|\) creates a V-shaped curve. This curve has a vertex at the origin (0,0) and symmetrically opens upwards. Each side of the V is a straight line with a slope of 1 on the right of the vertex and -1 on the left of the vertex. This shape is due to the absolute value taking the positive magnitude of any input.
The basic absolute value function illustrates fundamental properties of absolute value, such as symmetry and piecewise linearity.

graph transformations

Graph transformations involve changing the position, shape, or size of a graph by applying certain modifications to the function’s equation.
In the context of absolute value functions, we often modify the graph through vertical stretching or compressing by changing the coefficient in front of the absolute value. For example, in \(y = 2|x|\), the 2 causes the graph to stretch vertically, making it steeper.

• Vertical Stretch: When the coefficient > 1, such as in \(y = 2|x|\) or \(y = 4|x|\), the graph becomes steeper.
• Vertical Compression: When 0 < coefficient < 1, such as in \(y = \frac{1}{2}|x|\), the graph becomes less steep.

These transformations do not affect the vertex of the V, which remains at (0,0). Instead, they change the slopes of the lines that form the V.

graphing utility

A graphing utility, such as a graphing calculator or software like Desmos, is a powerful tool that assists in visualizing mathematical functions and their transformations.
Using a graphing utility makes it easier to compare different functions on the same axes. For example, you can graph \(y = |x|\), \(y = 2|x|\), and \(y = \frac{1}{2}|x|\) simultaneously to observe how changes in the coefficient affect the steepness of the graph. This visual aid helps identify patterns and understand the impact of coefficients on the graph of an absolute value function.
Moreover, graphing utilities often come with features that allow you to zoom in/out, adjust axes, and input specific points, making the learning process interactive and intuitive.

coefficients and steepness

The coefficient in front of the absolute value symbol \( |x| \) directly influences the steepness of the graph.

• If the coefficient is greater than 1 (e.g., \( y = 2|x| \) or \( y = 4|x| \)), the graph becomes steeper. This is because the y-values grow faster as the x-values increase.
• If the coefficient lies between 0 and 1 (e.g., \( y = \frac{1}{2}|x| \) or \( y = \frac{1}{4}|x| \)), the graph becomes less steep. The y-values increase at a slower rate compared to the x-values.
• If the coefficient equals 1, the graph has a standard steepness, with a slope of 1 on one side and -1 on the other.

This relationship allows you to predict the shape and steepness of the graph for any given absolute value function by just looking at the coefficient. For instance, for \( y = 5|x| \), the graph will be much steeper compared to \( y = 2|x| \). Conversely, for \( y = \frac{1}{4}|x| \), the graph will be flatter compared to \( y = \frac{1}{2}|x| \).