Problem 19 - Calculus with Concepts in Calculus

Question

Find an equation of the line described. Then sketch the line. The line with slope 3 and \(y\) intercept \(-3\)

Step-by-Step Solution

Answer

The equation of the line is \( y = 3x - 3 \).

Step 1: Understand Slope-Intercept Form

The equation of a line in the slope-intercept form is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Identify Given Values

From the problem statement, we know that the slope \( m = 3 \) and the y-intercept \( b = -3 \).

Step 3: Substitute Values into Equation

Replace \( m \) and \( b \) in the slope-intercept equation with the given values: \( y = 3x - 3 \).

Step 4: Sketch the Line

To sketch the line, start by plotting the y-intercept \( (0, -3) \) on the graph. From this point, use the slope to find another point by rising 3 units and running 1 unit to the right, which lands on \( (1, 0) \). Draw a line through these points.

Key Concepts

Equation of a Line

To understand the equation of a line, it's crucial to become familiar with the slope-intercept form: \( y = mx + b \). This equation is a way of expressing a straight line on a graph. Here, \( x \) and \( y \) represent the coordinates of any point on the line, \( m \) is the slope, and \( b \) is the y-intercept. The slope \( m \) shows how steep the line is, while the y-intercept \( b \) shows where the line crosses the y-axis.

In our example, we're dealing with the equation \( y = 3x - 3 \). This specific form allows us to quickly identify the line's characteristics. It provides a clear blueprint of how the line behaves and can be used to find any point on the line by plugging in a value for \( x \) and solving for \( y \). Understanding this concept is fundamental when graphing or predicting the behavior of lines.

Slope

The slope of a line is a measure that tells us how much the line inclines or declines as we move along it. In the slope-intercept equation \( y = mx + b \), the slope is represented by \( m \). The value of \( m \) informs us "how many units the line goes up or down" for every "one unit it goes to the right" on the graph.

For instance, a slope of 3 means that for every one unit we move to the right along the x-axis, we move three units up along the y-axis. This is often referred to as "rise over run."

• A positive slope indicates the line rises as it moves right.
• A negative slope would mean the line falls as it moves right.

Understanding slope helps us predict and graph the direction of a line.

Y-Intercept

The y-intercept is the point at which a graph intersects the y-axis. In the slope-intercept form, \( y = mx + b \), the parameter \( b \) stands for the y-intercept.

In simpler terms, this is the value of \( y \) when \( x \) equals zero. It's like saying "where does the line cross the y-axis?"

In our example equation \( y = 3x - 3 \), the y-intercept is -3, so the line crosses the y-axis at the point (0, -3). This gives us a specific point to start from when graphing a line.

Graphing Lines

Graphing a line using the slope-intercept form \( y = mx + b \) is straightforward once you grasp the equation. Here's how you can do it step by step.

Start by plotting the y-intercept. For example, in \( y = 3x - 3 \), the y-intercept is -3, so place a point at (0, -3) on the graph.

Next, use the slope to determine another point. Our slope, taken from \( y = 3x - 3 \), is 3, meaning we "rise" 3 units up for every 1 unit we "run" to the right. From (0, -3), move one to the right and three up to find a new point at (1, 0).

• Plot this second point.
• Draw a straight line through both points.

This method shows the line on the graph, matching the equation. With practice, this becomes intuitive and a fundamental skill in dealing with linear equations.