Problem 26 - College Algebra with Modeling and Visualization

Question

Exercises \(19-32:\) Graph the linear function by hand. Identify the slope and y-intercept. $$ f(x)=2 x-3 $$

Step-by-Step Solution

Answer

The slope is 2, and the y-intercept is -3; the line passes through points (0, -3) and (1, -1).

Step 1: Identify the Slope and y-intercept

The function given is in the form of the slope-intercept form, which is: \[ f(x) = mx + b \]In this case, the function is:\[ f(x) = 2x - 3 \]Comparing with the slope-intercept form, we identify the slope \( m = 2 \) and the y-intercept \( b = -3 \).

Step 2: Plot the y-intercept on the Graph

To begin graphing, first plot the y-intercept on the y-axis. The y-intercept is the point where the line crosses the y-axis, which corresponds to \( (0, -3) \). Place a point at \( (0, -3) \) on the graph.

Step 3: Use the Slope to Find Another Point

The slope \( m = 2 \) informs us that for every increase of 1 unit in \( x \), \( y \) increases by 2 units. Starting from the y-intercept \( (0, -3) \), move 1 unit to the right (x-direction) and 2 units up (y-direction) to find another point, which is \( (1, -1) \). Place this point on the graph.

Step 4: Draw the Line Through the Points

With both points \( (0, -3) \) and \( (1, -1) \) marked, use a ruler to draw a straight line through these points. Extend the line in both directions to display the linear function across the graph. Ensure that the line extends fully through both points and follows the pattern of the slope.

Step 5: Verify the Line with Additional Points

Check your graph by choosing another x-value to ensure it lies on the line. For example, use \( x = 2 \):\[ f(x) = 2(2) - 3 = 4 - 3 = 1 \]The point \( (2, 1) \) should also be on the line you drew. Plot this point to confirm your line is accurate.

Key Concepts

Graphing Linear Equations

Graphing linear equations is an essential skill in algebra that allows us to visually represent relationships between variables. A linear equation typically forms a straight line on a graph, which you can identify by its constant rate of change, or slope. To graph a linear equation, you'll need to determine its slope and y-intercept, then use these to plot points and draw the line.

The linear equation provided in the example is in the form of a function: \( f(x) = 2x - 3 \). This equation tells us how the variable \( y \) (or \( f(x) \)) changes with \( x \). Begin by identifying key components of the equation that will guide the graphing process.

• Start by locating the y-intercept and plotting it on the y-axis.
• Use the slope to identify another point on the graph.
• Draw a line through these points to complete the graph.

By following these steps, you can accurately graph linear equations, making them much easier to understand and interpret.

Slope-Intercept Form

The slope-intercept form is a specific way to represent linear equations, making it easier to graph them efficiently. This form is expressed as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept.

In our example, \( f(x) = 2x - 3 \) matches this form perfectly. Here, the slope \( m = 2 \) indicates the line rises 2 units for every 1 unit increase in \( x \). Meanwhile, the y-intercept \( b = -3 \) is the point where the line crosses the y-axis.

• Slope \( (m) \): This tells us how steep the line is. A higher value means a steeper incline.
• Y-intercept \( (b) \): This is the starting point of the line on the y-axis when \( x = 0 \).

Understanding this form allows you to quickly graph lines by identifying the slope and y-intercept, providing a straightforward path towards visualizing linear relationships.

Plotting Points

Plotting points is a fundamental step in graphing that helps establish the precise location of a line on a coordinate plane. When starting to graph, it's crucial to accurately plot key points such as the y-intercept and additional points determined by the slope.

For our given equation \( f(x) = 2x - 3 \), begin by plotting the y-intercept at \( (0, -3) \). This point is crucial as it serves as an anchor for the line. Next, use the slope to find another point. The slope \( m = 2 \) tells us that from \( (0, -3) \), if you move 1 unit to the right (along the x-axis), you should move 2 units up (along the y-axis) to reach the new point \( (1, -1) \).

• Ensuring Accuracy: Check your plotted points with additional values if needed, like \( (2, 1) \) in our example, to ensure the line's accuracy.
• Visual Aid: Plotted points work like a connect-the-dots that guide the drawing of the line.

When points are plotted with care, you create a clear and precise graph of the linear equation, helping to visualize how changes in \( x \) affect \( y \).