Problem 79 - College Algebra with Corequisite Support

Question

For the following exercises, sketch the graph of each equation. $$ f(x)=\frac{2}{3} x-3 $$

Step-by-Step Solution

Answer

Graph the line through (0, -3) and (3, -1) with a slope of \(2/3\).

Step 1: Identify the slope and intercept

The given equation is in the slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For the equation \( f(x) = \frac{2}{3} x - 3 \), the slope \( m \) is \( \frac{2}{3} \), and the y-intercept \( b \) is \( -3 \).

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 this equation, plot the point (0, -3) on the graph.

Step 3: Use the slope to plot a second point

The slope \( \frac{2}{3} \) means that for every 3 units you move to the right on the x-axis, you move up 2 units on the y-axis. Starting from the y-intercept point (0, -3), move 3 units to the right, reaching x = 3, and 2 units up, reaching y = -1. Plot this second point (3, -1) on the graph.

Step 4: Draw the line

Using a ruler, draw a straight line through the points (0, -3) and (3, -1). Extend the line across the grid in both directions, keeping the same slope. This line represents the graph of the equation \( f(x) = \frac{2}{3}x - 3 \).

Key Concepts

Slope-Intercept Form

When you're graphing linear equations, the slope-intercept form is incredibly helpful. It's expressed as \(y = mx + b\). In this equation, \(m\) represents the slope, and \(b\) is the y-intercept. By writing an equation in this form, you can easily identify key features of a line and graph it quickly.

**Why use it?**

• Simple graphing: It allows for quick plotting of the y-intercept first.
• Clarity: Easily shows the slope, helping in understanding how steep a line is.

For example, in \(f(x) = \frac{2}{3}x - 3\), the slope \(m\) is \(\frac{2}{3}\) and the y-intercept \(b\) is \(-3\). This makes our job of sketching the graph straightforward and effective.

Slope

The slope of a line, represented by \(m\) in the slope-intercept formula, indicates how slanted a line is. It tells you how much the line rises or falls vertically for every unit it moves horizontally. The slope is often called "rise over run."

**Understanding the rise over run**

• Positive slope: The line goes upwards from left to right.
• Negative slope: The line goes downwards from left to right.
• Zero slope: The line is horizontal, showing no change.
• Undefined slope: Indicates a vertical line.

In our example \(f(x) = \frac{2}{3}x - 3\), with a slope of \(\frac{2}{3}\), for every 3 units you move to the right, you move up by 2 units.
Using the slope, start from the y-intercept and find a second point by applying this rise and run. This helps in sketching the graph accurately.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis. It's easily found in the slope-intercept form as the constant \(b\). This point provides you with a starting position on the graph.

**Characteristics of the y-intercept:**

• It's the value of \(y\) when \(x=0\).
• A critical reference to begin drawing a line.

In the equation \(f(x) = \frac{2}{3}x - 3\), the y-intercept is \(-3\). This means the first point to plot on your graph is \((0, -3)\).
From this starting point, use the slope to find additional points, ensuring you capture the line's correct trajectory and steepness. Placing the y-intercept effectively simplifies the process of graphing linear equations.