Problem 42 - Precalculus Mathematics for Calculus

Question

Use a graphing device to graph the given family of lines in the same viewing rectangle. What do the lines have in common? $$y=m x-3 \quad \text { for } m=0, \pm 0.25, \pm 0.75, \pm 1.5$$

Step-by-Step Solution

Answer

All lines have the same y-intercept at \(y = -3\).

Step 1: Understanding the Exercise

The problem asks us to graph several linear functions with the equation form \(y = mx - 3\) for different values of \(m\). These functions represent a family of lines with a common y-intercept at -3.

Step 2: Identify the Values of the Slope \(m\)

The given slopes for the lines are \(m = 0\), \(m = \pm 0.25\), \(m = \pm 0.75\), and \(m = \pm 1.5\). These values will be used to generate different lines.

Step 3: Graphing the Lines

Using a graphing tool, plot each linear equation by replacing \(m\) with the given values. Start with \(y = 0*x - 3\) which is a horizontal line at \(y = -3\), and then plot lines for \(m = 0.25\), \(m = -0.25\), \(m = 0.75\), \(m = -0.75\), \(m = 1.5\), and \(m = -1.5\). Make sure all graphs are in the same viewing rectangle for comparison.

Step 4: Analyze the Graph

Each line intersects the y-axis at \(y = -3\). The only difference between these lines is their slopes, causing them to have different inclines but share the same y-intercept.

Key Concepts

Slope-Intercept Form

When graphing linear equations, the slope-intercept form is a very helpful format. A linear equation in the slope-intercept form is written as:

• \( y = mx + b \)

where:

• \( m \) is the slope and tells us how steep the line is.
• \( b \) is the y-intercept, indicating where the line crosses the y-axis.

In our specific exercise, the equations are in the form \( y = mx - 3 \), making it clear that all lines in the family have the same y-intercept \( b = -3 \). This means no matter how the slope \( m \) changes, each line will cross the y-axis at the same point, simplifying the process of graphing them on the same set of axes with shared features.

Family of Lines

In mathematical terms, a family of lines refers to a group of lines with a common characteristic. For the exercise, all lines share a y-intercept at \( y = -3 \). This means they all cross the y-axis at this point but diverge from each other at different angles based on their slope \( m \).

Different slopes represent different inclines:

• A line with a slope \( m = 0 \) is perfectly horizontal, indicated as \( y = -3 \).
• Positive slopes like \( m = 0.25 \), \( m = 0.75 \), and \( m = 1.5 \) rise upwards as you move from left to right, showing an increasing relationship between \( x \) and \( y \).
• Negative slopes such as \( m = -0.25 \), \( m = -0.75 \), and \( m = -1.5 \) fall downwards as \( x \) increases, showing a decreasing trend.

Understanding this family helps us see how a single feature, like the y-intercept, can tie multiple lines together, while their direction is shaped by the variation in slopes.

Graphing Tools

Graphing tools are crucial when visualizing linear equations, especially when dealing with multiple lines. These can include software like graphing calculators, computer applications, or even online graphing tools.

Here's what they help accomplish:

• Precise plotting of lines by calculating values based on equations.
• Allowing for easy comparison of multiple lines by displaying them in the same coordinate plane or viewing rectangle.
• Helping spot patterns or differences, such as the shared y-intercept or varying slopes in lines.

To graph the exercise lines, insert each equation \( y = mx - 3 \) into your graphing tool by varying \( m \) as needed. Confirm each line appears together for clear interpretation of their similarities and differences. Such tools facilitate understanding how linear equations behave within a family and can enhance your learning experience by instantly showing results. They serve as a visual aid making abstract mathematical concepts more tangible.