Problem 44 - Precalculus Essentials

Question

Give the slope and \(y\)-intercept of each line whose equation is given. Then graph the linear function. $$f(x)=\frac{3}{4} x-3$$

Step-by-Step Solution

Answer

The slope of the line is \(\frac{3}{4}\) and the y-intercept is -3. The graph of the function is a straight line that passes through the points (0,-3) and (4,0).

Step 1: Identify the Slope and y-intercept

From the given function \(f(x)=\frac{3}{4}x - 3\), we can see that the slope (m) is \(\frac{3}{4}\) and the y-intercept (c) is -3.

Step 2: Plot the y-intercept

Start by plotting the y-intercept on a graph. The y-intercept is the point where the line crosses the y-axis, so this point is (0,-3).

Step 3: Use the Slope to Find Another Point

The slope is the rise over the run (change in y over change in x). Because our slope is \(\frac{3}{4}\), this means we rise 3 units and run 4 units. Starting from the y-intercept (0,-3), we can move up 3 units and to the right 4 units to find another point on the line, which would be (4,0).

Step 4: Draw the Line

Lastly, we draw a line through the points we plotted to create the graph of the function. This line represents all the possible solutions for the function \(f(x)=\frac{3}{4}x - 3\)

Key Concepts

Understanding Slope

When analyzing a linear equation, one of the most important aspects to understand is the slope. The slope is a measure of how steep a line is. In the equation of a line in the form of \( y = mx + c \), \( m \) represents the slope.
For example, in the given function \( f(x) = \frac{3}{4}x - 3 \), the slope is \( \frac{3}{4} \). This value tells us two things:

• The line rises by 3 units for every 4 units it moves horizontally. This is often described as "rise over run."
• A positive slope indicates the line is rising as it moves from left to right. If the slope were negative, the line would fall as you move from left to right.

Recognizing and calculating the slope helps us understand the direction and steepness of the line, which are crucial for accurately graphing the equation.

Breaking Down the Y-intercept

The y-intercept is where the line crosses the y-axis on a graph. It is represented by \( c \) in the linear equation \( y = mx + c \). For our equation \( f(x) = \frac{3}{4}x - 3 \), the y-intercept is -3.

• This means the line will cross the y-axis at the point (0, -3).
• To find the y-intercept, substitute 0 for \( x \) in the equation, which gives you the y-value where the line intersects the y-axis.

The y-intercept is a crucial starting point for drawing the graph of a line, as it is one of the key reference points to plot before using the slope to find additional points.

Graphing Linear Equations Made Simple

Graphing linear equations involves plotting points and drawing a line through these points. Here’s how you can do it using the slope and y-intercept:

• First, identify and plot the y-intercept on the graph. For \( f(x) = \frac{3}{4}x - 3 \), the y-intercept is -3, so you plot the point (0, -3).
• Next, use the slope to find your second point. From (0, -3), move up 3 units (because of the "rise") and to the right 4 units (because of the "run"). This places your second point at (4,0).
• Draw a straight line through these two points, extending it across the graph. This line represents all solutions to the equation \( f(x) = \frac{3}{4}x - 3 \).

By understanding and applying the concepts of slope and y-intercept, graphing linear equations becomes a straightforward process. It provides a visual representation of how the equation behaves.