Problem 15 - Brief Applied Calculus

Question

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. \(y=3 x-4\)

Step-by-Step Solution

Answer

Slope \(m = 3\); y-intercept is \((0, -4)\); graph is a straight line.

Step 1: Identify the equation format

The equation is given in the slope-intercept form: \( y = mx + b \), which makes it easy to identify the slope and y-intercept.

Step 2: Determine the slope \(m\)

In the equation \( y = 3x - 4 \), the coefficient of \( x \) is the slope \( m \). Therefore, \( m = 3 \).

Step 3: Find the y-intercept \(b\)

The y-intercept \( b \) is the constant term in the equation. In this case, the equation is \( y = 3x - 4 \), so the y-intercept \( b = -4 \).

Step 4: Plot the y-intercept on the graph

Begin by plotting the y-intercept \((0, -4)\) on the graph. This is the point where the line crosses the y-axis.

Step 5: Use the slope to draw the line

With a slope \( m = 3 \), this means that for every 1 unit you move to the right on the x-axis, you move up 3 units on the y-axis. From the point \((0, -4)\), move one unit to the right to \(x = 1\), then up three units to \(y = -1\) and plot this second point.

Step 6: Draw the line

Using a ruler, connect the y-intercept and the second point \((1, -1)\) with a straight line and extend it in both directions. This line represents the graph of the equation \( y = 3x - 4 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward way to express a linear equation. A linear equation in this form is written as \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept. This format is incredibly useful because it allows you to immediately identify these two key features of a line just by looking at the equation itself.

This approach simplifies both the understanding and graphing of a line. Using the slope-intercept form, you can quickly determine how the line behaves by recognizing the rate at which it rises or falls (the slope) and where it crosses the y-axis (the y-intercept).

Always remember:

• \( m \) (slope) indicates the steepness and direction of the line.
• \( b \) (y-intercept) is the point where the line crosses the y-axis.

Understanding how to use this form of the equation is a crucial step in mastering linear equations.

Slope Calculation

Calculating the slope of a line requires understanding how much the line rises or falls for each unit it moves horizontally. This change is visually expressed as 'rise over run.' In a linear equation in slope-intercept form, the slope is simply the coefficient in front of the \( x \) term.

When working with an equation like \( y = 3x - 4 \), finding the slope is easy:

• Identify the coefficient of \( x \). This number is the slope \( m \).
• In our example, the slope \( m = 3 \).

This means for each step you move to the right along the x-axis, you move up 3 steps along the y-axis. A positive slope like \( 3 \) means the line ascends as it moves to the right. Understanding how to calculate the slope helps in graphing the line accurately.

Y-Intercept Identification

Y-intercept identification is crucial for graphing a linear equation accurately. It gives us the starting point on the y-axis where the graph of the equation will begin.

The y-intercept in a slope-intercept form equation \( y = mx + b \) is the constant \( b \). It tells us the exact point where the line crosses the y-axis. In our example, \( y = 3x - 4 \), the y-intercept \( b \) is \( -4 \).

This means the line crosses the y-axis at the point \( (0, -4) \). When plotting the graph, you start by placing a point at this coordinate on the y-axis, and then use the slope to determine the direction and steepness of the line.

Effectively identifying the y-intercept is essential, especially as it serves as the foundation for graphing the rest of the line accurately. Once the y-intercept is plotted, the slope guides you in plotting additional points.