Problem 14 - College Algebra with Modeling and Visualization

Question

Graph \(y=f(x)\) by hand by first plotting points to determine the shape of the graph. $$ f(x)=|0.5 x| $$

Step-by-Step Solution

Answer

Plot points to reveal a V-shape graph, symmetric around the y-axis, with the vertex at the origin.

Step 1: Understanding the Function

The given function is a linear absolute value function, expressed as \( f(x) = |0.5x| \), which means for any input \(x\), the output \(f(x)\) is the absolute value of half of \(x\). This function will create a V-shaped graph.

Step 2: Creating a Table of Values

Select several values of \(x\) to evaluate \(f(x)\). Choose both positive and negative values to see the symmetry:- \( x = -4 \), \( f(-4) = |0.5(-4)| = 2 \)- \( x = -2 \), \( f(-2) = |0.5(-2)| = 1 \)- \( x = 0 \), \( f(0) = |0.5(0)| = 0 \)- \( x = 2 \), \( f(2) = |0.5(2)| = 1 \)- \( x = 4 \), \( f(4) = |0.5(4)| = 2 \)

Step 3: Plotting Points

Plot these points on the coordinate plane: - \((-4, 2)\)- \((-2, 1)\)- \((0, 0)\)- \((2, 1)\)- \((4, 2)\).Ensure the scale of the axes is appropriate to clearly see the shape.

Step 4: Connecting the Points

Draw straight lines connecting the points from left to right. The graph will show a V-shape with the vertex at \( (0, 0) \), illustrating how the absolute value operation affects the function. The lines should be symmetrical to the y-axis.

Step 5: Analyzing the Graph

Observe that the graph consists of two straight lines that meet at the origin, \((0,0)\). The line from negative to zero has a positive slope, while the line from zero to positive x-values also has a positive slope. This reflects the symmetry characteristic of absolute value functions.

Key Concepts

Plotting Points

Understanding how to plot points is crucial for graphing functions. It's like building a house—each point is a brick in your mathematical construction. For the function \(f(x) = |0.5x|\), we start by selecting various \(x\) values from the number line. Choosing a range that includes both positive and negative values, such as -4, -2, 0, 2, and 4, is essential because it reveals the symmetry of the graph.
To plot a point:

• First, calculate \(f(x)\). For instance, if \(x = -4\), then \(f(-4) = |0.5(-4)| = 2\).
• Next, pair this \(f(x)\) value with your chosen \(x\) value, creating a coordinate point, like (-4, 2).
• Repeat this process for all selected \(x\) values to generate enough points for your graph.

Plotting these points on the graph helps illustrate the function's shape, turning abstract numbers into a visual part of a bigger picture.

Coordinate Plane

The coordinate plane is your canvas when graphing functions. It consists of two number lines that intersect at right angles: the x-axis (horizontal) and the y-axis (vertical). Every point on the plane is denoted as an ordered pair \((x, y)\), where \(x\) indicates the position along the x-axis and \(y\) along the y-axis.
For our function \(f(x) = |0.5x|\), the coordinate plane helps to visualize how changing \(x\) affects \(f(x)\). Once you plot your points—such as (-4, 2), (-2, 1), (0, 0), (2, 1), (4, 2)—you place each one at its corresponding position on the plane. Here are some quick tips to remember:

• Always ensure each axis is properly labeled and evenly spaced.
• Include both positive and negative directions to capture the complete picture of the function.
• The origin \((0, 0)\) should be correctly identified, as it often serves as a point of symmetry for graphs.

By accurately plotting these points, you'll see the absolute value function's distinctive V-shape emerge on the plane.

Linear Functions

Linear functions form the backbone of many mathematical concepts, and they show up prominently when graphing absolute value functions. These are typically written in the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In the case of absolute value functions, though, the straight line is affected by the absolute operation, resulting in a distinctive V-shape instead of a single continuous line.
The function \(f(x) = |0.5x|\) generates two linear pieces: each side of the V-shape is a line with a slope of 0.5, but they diverge in opposite directions from the vertex point \((0, 0)\). Both parts:

• Exhibit symmetry—one rising from negative x-values to the origin and the other ascending from the origin to positive x-values.
• Illustrate how absolute value affects linear functions, not crossing below the x-axis since all outputs are non-negative.
• Have consistent and equal slopes on either side of the vertex, reflecting the absolute value symmetry.

By understanding how linear functions are integrated into the structure of absolute value functions, you can better grasp the nature of their graphs and the visual representation of mathematical relationships.