Problem 25 - College Algebra with Modeling and Visualization

Question

Exercises \(19-32:\) Graph the linear function by hand. Identify the slope and y-intercept. $$ f(x)=4-\frac{1}{2} x $$

Step-by-Step Solution

Answer

Slope: -\frac{1}{2}, y-intercept: 4. Graph by plotting (0, 4) and (2, 3) and drawing a line through them.

Step 1: Identify the standard form

The given linear function is \( f(x) = 4 - \frac{1}{2} x \). Re-arrange it into the standard slope-intercept form, \( y = mx + b \). Rewrite the function as \( y = -\frac{1}{2}x + 4 \), where \( m = -\frac{1}{2} \) is the slope, and \( b = 4 \) is the y-intercept.

Step 2: Plot the y-intercept

The y-intercept is the point where the line crosses the y-axis. From the equation \( y = -\frac{1}{2}x + 4 \), the y-intercept is \( b = 4 \). Plot the point \( (0, 4) \) on the graph.

Step 3: Use the slope to find another point

The slope \( m = -\frac{1}{2} \) means that for every 1 unit increase in \( x \), \( y \) decreases by \( \frac{1}{2} \) units. Starting from the y-intercept \( (0, 4) \), move 2 units to the right to \( x = 2 \), then 1 unit down to \( y = 3 \), plotting the point \( (2, 3) \).

Step 4: Draw the line

Use a ruler to draw a straight line through the two points \( (0, 4) \) and \( (2, 3) \) that you plotted. Extend the line across the graph, ensuring it goes through both points.

Key Concepts

Graphing Linear Equations

Graphing linear equations is an essential math skill that allows us to visually interpret the solutions to an equation. Linear equations, like the one in this exercise, form straight lines when plotted on a graph. To graph a linear equation, we need to understand its general form:

• Linear equations can be written in the form \( y = mx + b \).
• Here, \( m \) represents the slope, and \( b \) is the y-intercept.

The slope indicates the steepness of the line, and the y-intercept represents the point where the line crosses the y-axis. By identifying these two key components, we can plot the graph accurately. These are vital for establishing the line's direction and position on a coordinate plane.

Slope-Intercept Form

The slope-intercept form of a linear equation is a powerful tool for graphing. The format \( y = mx + b \) makes it easy to extract the slope and y-intercept directly from the equation.

• The slope \( m \) tells us how to move from one point on the line to another.
• A positive slope means the line inclines upwards, while a negative slope indicates a downward slope.
• The y-intercept \( b \) is simply where \( y \) is intersected when \( x = 0 \).

In our exercise, after rearranging, we have \( y = -\frac{1}{2}x + 4 \). Thus, the slope is \(-\frac{1}{2}\) and the y-intercept is \(4\). This form allows us to quickly identify these crucial points without additional calculations.

Plotting Points on a Graph

Plotting points on a graph is straightforward once the slope and y-intercept are known. The starting point is often the y-intercept. For the equation \( y = -\frac{1}{2}x + 4 \), begin by plotting \( (0, 4) \) on the y-axis. Next, use the slope to find other points. Since the slope is \(-\frac{1}{2}\):

• Move 2 units to the right on the x-axis.
• Then, move 1 unit down on the y-axis due to the negative sign which indicates a decrease in \( y \).

This brings us to the point \( (2, 3) \). Connecting these points with a ruler will extend the line across the graph, illustrating the linear equation clearly. Visualizing these steps can solidify one's understanding of how linear equations behave graphically.