Problem 21 - College Algebra with Modeling and Visualization

Question

Exercises \(19-32:\) Graph the linear function by hand. Identify the slope and y-intercept. $$ f(x)=\frac{1}{2} x-2 $$

Step-by-Step Solution

Answer

Slope: \( \frac{1}{2} \), y-intercept: -2; Graph through (0, -2) and (2, -1).

Step 1: Identify the Slope and y-Intercept

For the linear function \( f(x) = \frac{1}{2}x - 2 \), it is in the slope-intercept form \( y = mx + b \). Here, \( m = \frac{1}{2} \) is the slope, and \( b = -2 \) is the y-intercept.

Step 2: Plot the y-Intercept

Locate the y-intercept on the graph. For \( f(x) = \frac{1}{2}x - 2 \), the y-intercept is at the point (0, -2). Plot this point on the y-axis.

Step 3: Use the Slope to Find Another Point

From the y-intercept (0, -2), use the slope \( \frac{1}{2} \) to find another point. The slope means 'rise over run'; from (0, -2), move up 1 unit and right 2 units to reach the point (2, -1). Plot this point.

Step 4: Draw the Line

With the points (0, -2) and (2, -1) plotted, use a ruler to draw a straight line through these points. This line represents the graph of the function \( f(x) = \frac{1}{2}x - 2 \).

Key Concepts

Slope

In a linear function, the slope represents how steep the line is. It's a measure of change, specifically "rise over run." In the equation \( y = mx + b \), the slope \( m \) is the coefficient of \( x \). It's crucial because it tells us how to move from one point to another along the line. For the function \( f(x) = \frac{1}{2}x - 2 \), the slope \( m \) is \( \frac{1}{2} \). This means that for every 2 units you move horizontally to the right, you move 1 unit up.
Understanding slope is about recognizing this consistent rate of change.
The slope can be positive, negative, zero, or undefined:

• Positive slope: line rises as it moves to the right.
• Negative slope: line falls as it moves to the right.
• Zero slope: line is horizontal.
• Undefined slope: line is vertical.

In our example, \( \frac{1}{2} \) is a positive slope, so the line rises gently upwards.

Y-Intercept

The y-intercept of a line is where it crosses the y-axis. It's represented by \( b \) in the linear equation \( y = mx + b \). This point is vital because it gives us a starting location for graphing our line. In the function \( f(x) = \frac{1}{2}x - 2 \), the y-intercept \( b \) is equal to -2.
So, the line crosses the y-axis at the point (0, -2).
The y-intercept is the point where the value of \( x \) is 0. By plotting this point first, you establish a reference from which you can use the slope to plot additional points.
Often, students find visualizing the y-intercept straightforward as it is a definite position where any line drawn would strike the vertical axis of the graph.

Graphing

Graphing a linear function involves accurately plotting its points on a Cartesian plane and drawing a line through them. The most intuitive approach starts with plotting the y-intercept, then uses the slope to find other points. Let's go through the function \( f(x) = \frac{1}{2}x - 2 \):

• Step 1: Start with the y-intercept at (0, -2). Plot this on the y-axis.
• Step 2: Use the slope \( \frac{1}{2} \): from (0, -2), move up 1 unit (the "rise") and right 2 units (the "run") to locate the next point at (2, -1).
• Step 3: Plot the second point (2, -1).
• Step 4: Use a ruler to draw a straight line through both points.

As simple as it sounds, precision in these steps ensures accuracy. By connecting these points correctly, you visually represent the linear relationship defined by your equation. Graphing not only provides a visual aspect but also strengthens the understanding of how algebraic changes affect visual output.