Problem 30 - College Algebra with Modeling and Visualization

Question

Exercises \(19-32:\) Graph the linear function by hand. Identify the slope and y-intercept. $$ g(x)=\frac{3}{4} x-2 $$

Step-by-Step Solution

Answer

Slope is \( \frac{3}{4} \) and y-intercept is -2. Graph passes through (0, -2) and (4, 1).

Step 1: Identify the Slope and Y-intercept

The given linear function is \( g(x) = \frac{3}{4}x - 2 \). This equation is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, the slope \( m \) is \( \frac{3}{4} \) and the y-intercept \( b \) is -2.

Step 2: Plot the Y-intercept

Start by plotting the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis. For \( g(x) = \frac{3}{4} x - 2 \), the y-intercept is -2. So, plot the point (0, -2) on the graph.

Step 3: Use the Slope to Find Another Point

The slope \( \frac{3}{4} \) means that for every 3 units the line rises, it runs 4 units to the right. From the y-intercept (0, -2), move up 3 units and 4 units to the right to find another point: (4, 1). Plot this point on the graph.

Step 4: Draw the Line

Using the two points plotted—(0, -2) and (4, 1)—draw a straight line through these points. Extend the line across the graph in both directions to complete the graph of the linear function.

Key Concepts

Slope-Intercept Form

When graphing linear functions, the slope-intercept form is a handy equation. It's written as \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept. This form provides valuable insights into the behavior of a line on a graph.
The slope \( m \) indicates the steepness and direction of the line. A positive slope means the line rises, while a negative slope means it falls. Finding the y-intercept \( b \) is also straightforward. It's the point where the line crosses the y-axis.
In the given function \( g(x) = \frac{3}{4}x - 2 \), the slope is \( \frac{3}{4} \) and the y-intercept is -2. This means the line rises 3 units for every 4 units it moves to the right, crossing the y-axis at -2.

• The form \( y = mx + b \) helps you quickly identify these elements.
• Slope tells you how to move from point to point on the graph.
• The y-intercept gives you a starting point on the y-axis.

Y-Intercept

The y-intercept is a crucial part of understanding linear functions. It provides the starting point on the graph from which the line will extend. In the equation \( y = mx + b \), the y-intercept is represented by \( b \).
For the function \( g(x) = \frac{3}{4}x - 2 \), the y-intercept is -2. This tells you that the line intersects the y-axis at the point (0, -2).
Knowing the y-intercept is useful because it sets a concrete point from where you can begin plotting the rest of your graph.

• Start by plotting the y-intercept, (0, -2) in this case, on the graph.
• This point confirms where the line crosses the y-axis.
• It acts as a guide to ensuring accuracy when drawing your line.

Plotting Points

Plotting points is an essential skill when graphing linear functions by hand. Once you have your y-intercept and slope, you can start adding points to the graph. This helps to visualize the line the equation represents.
Begin with the y-intercept point you’ve marked—say, (0, -2). Use the slope \( \frac{3}{4} \) to find another point. This slope signals a rise of 3 units for every run of 4 units to the right. From (0, -2), move up 3 units and 4 units to the right, reaching the point (4, 1).
By connecting these points, you can then draw a straight line, completing your graph.

• The slope helps determine how to travel from one point to another.
• The points you plot form the backbone of your line graph.
• Ensure accuracy by carefully counting the rise and run according to the slope.