Problem 33 - A Graphical Approach to Precalculus with Limits

Question

Graph each equation by hand. $$y=\frac{1}{2} x, y=\left|\frac{1}{2} x\right|$$

Step-by-Step Solution

Answer

The graph of \( y = \frac{1}{2}x \) is a straight line and \( y = \left| \frac{1}{2}x \right| \) forms a V-shape.

Step 1: Understand the Equation

We have two equations to graph: the first is a linear function, \( y = \frac{1}{2}x \), and the second is a piecewise function, \( y = \left| \frac{1}{2}x \right| \). The goal is to plot these on the same set of axes.

Step 2: Graph the Equation \( y = \frac{1}{2}x \)

Start by identifying the y-intercept and slope. The y-intercept of \( y = \frac{1}{2}x \) is 0, meaning the line passes through the origin (0,0). The slope is \( \frac{1}{2} \), indicating that for every 2 units increase in x, y increases by 1 unit.

Step 3: Plot Points for \( y = \frac{1}{2}x \)

Choose a couple of x-values to plot: for example, when \( x = 2, y = 1 \); when \( x = 4, y = 2 \). Similarly, for negative values: when \( x = -2, y = -1 \); when \( x = -4, y = -2 \). Join these with a straight line passing through the origin.

Step 4: Graph \( y = \left| \frac{1}{2}x \right| \)

Since this is an absolute value function, the graph reflects the negative part of \( y = \frac{1}{2}x \) above the x-axis. The equation is divided into two cases: when \( x \geq 0 \), it behaves as \( y = \frac{1}{2}x \), and when \( x < 0 \), \( y = -\frac{1}{2}x \).

Step 5: Plot Points for \( y = \left| \frac{1}{2}x \right| \)

For \( x \geq 0 \), use points identified in Step 3. For \( x < 0 \), reflect the negative points above the x-axis: when \( x = -2, y = 1 \); when \( x = -4, y = 2 \). This forms a V-shape with the vertex at the origin.

Step 6: Analyze the Graph

The graph of \( y = \frac{1}{2}x \) is a straight line through the origin, and the graph of \( y = \left| \frac{1}{2}x \right| \) is a V-shape also starting at the origin and opening upwards.

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions you will encounter in mathematics. They are called "linear" because they graph as straight lines. The general form for a linear equation is \( y = mx + b \), where:

• \( m \) is the slope of the line
• \( b \) is the y-intercept, the point where the line crosses the y-axis

Understanding how to graph linear functions involves identifying these two key components. In the example given, the equation \( y = \frac{1}{2}x \) has a slope of \( \frac{1}{2} \) and a y-intercept of 0, meaning the line passes through the origin.

To graph a linear function, you can:

• Start from the y-intercept on the graph.
• Use the slope to determine the rise over run (rise/run) to plot another point.
• Draw a straight line through these points to extend it in both directions.

This helps in visualizing how the line covers the coordinate plane.

Absolute Value Functions

Absolute value functions introduce a unique trait compared to regular linear functions. They hinge on the concept of absolute value, which measures the distance of a number from zero on a number line, always as a non-negative number.

For graphing purposes, the absolute value function \( y = \left| \frac{1}{2}x \right| \) alters the original line \( y = \frac{1}{2}x \) by reflecting the negative values of \( y \) to positive ones. Therefore, this graph forms a V-shape with:

• The vertex at the origin (0,0).
• Each arm of the V extends in the opposite direction symmetrically away from the y-axis.

The absolute value function is essentially a piecewise function, formed by two linear portions:

• For \( x \geq 0 \), it follows the line \( y = \frac{1}{2}x \).
• For \( x < 0 \), it mirrors this line as \( y = -\frac{1}{2}x \).

By plotting points as in the linear case and reflecting negative outcomes about the x-axis, you can clearly see the hallmark V-like structure.

Piecewise Functions

Piecewise functions are fascinating as they comprise different "pieces" of function segments, each defined over specific intervals of the domain. They allow a single function to exhibit multiple behaviors within its graph.

In the case of the function \( y = \left| \frac{1}{2}x \right| \), it is a specific type of piecewise function due to its two linear segments:

• \( y = \frac{1}{2}x \) for \( x \geq 0 \): This part manifests like the original line.
• \( y = -\frac{1}{2}x \) for \( x < 0 \): This reflects the line indicating how piecewise functions can "switch" their behavior at certain points.

Graphing this function effectively requires discerning these intervals and behavior changes.

The beauty of piecewise functions lies in their versatility in modeling real-world scenarios where different situations demand different rules or equations, creating a seamless and continuous graph that adjusts its posture according to the input.