Problem 37 - Precalculus

Question

(a) Use a graphing utility to graph the following three parallel lines in the standard viewing rectangle: \(y+4=-0.5(x-2) ; y-3=-0.5(x+2) ; y=-0.5 x\) (b) Experiment with different settings for \(\mathrm{Xmin}, \mathrm{Xmax}\) Ymin, and Ymax. In each case, do the three lines still appear to be parallel?

Step-by-Step Solution

Answer

The lines remain parallel across different viewing settings.

Step 1: Rewrite the Equations

Begin by rewriting each given equation in the form of \(y = mx + b\) to identify the slope and intercept clearly.\(y + 4 = -0.5(x - 2)\) simplifies to \(y = -0.5x + 5\). \(y - 3 = -0.5(x + 2)\) simplifies to \(y = -0.5x - 1\). The third equation is already in slope-intercept form: \(y = -0.5x\).

Step 2: Identify Key Features

For each equation, observe that the slope \(m\) is consistent across all three: \(-0.5\). This consistency means the lines are parallel. Their y-intercepts are different: \(5\), \(-1\), and \(0\).

Step 3: Plot the Equations Using a Graphing Tool

Utilize a graphing utility to plot each of these lines in the standard viewing window (usually \([-10, 10]\) for both x and y axes). Each line should appear parallel to the others because they share the same slope \(-0.5\).

Step 4: Adjust Viewing Window Settings

Experiment by adjusting Xmin, Xmax, Ymin, and Ymax values in your graphing tool. Try different ranges, such as Xmin = -20, Xmax = 20, Ymin = -15, Ymax = 15. Observe whether the lines remain parallel visually across different settings.

Step 5: Conclusion: Parallelism in Different Windows

Regardless of the changes in the graph view settings, the lines should still appear parallel. Parallel lines are characterized by having the same slope, which is unaffected by the graph's viewing window.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a simple yet powerful way to understand how a line behaves on a graph. Any straight line can be represented in this form as \(y = mx + b\). Here,

• \(m\) is the slope, which indicates the steepness of the line. It tells us how much the y-value increases or decreases for each increase of one unit in x.
• \(b\) is the y-intercept. This point on the y-axis is where the line crosses when x equals zero.

Essentially, the slope determines the direction:

• A positive slope means the line rises as it moves to the right.
• A negative slope means it falls.
• A slope of zero indicates a horizontal line.

With the three equations in the exercise, we rewrote them into the slope-intercept form:- \(y = -0.5x + 5\)- \(y = -0.5x - 1\)- \(y = -0.5x\)This transform helps us to quickly see that all lines share the same slope of -0.5, meaning they are parallel.

Graphing Utilities

Graphing utilities are awesome tools that assist students in visualizing mathematical equations. Using such utilities, you can plot lines with their exact equations as input. Here's how you can utilize them:

• Input the equations in slope-intercept form, which makes them easy for the graphing tool to interpret. This is why rewriting the equations is key.
• Visualize multiple lines at once, which allows you to compare their properties like slope and intercept.
• Adjust settings like xmin, xmax, ymin, and ymax to see different parts of the grid.

For example, by plotting the equations \(y = -0.5x + 5\), \(y = -0.5x - 1\), and \(y = -0.5x\), the graphing utility shows us three lines aligned with the same slope. No matter how you change the viewing window settings, such as expanding the axes to x ranging from -20 to 20 or y from -15 to 15, these lines remain visually parallel. This fact beautifully illustrates the consistency of parallel lines in graphical displays.

Linear Equations

Linear equations are the essence of algebra when dealing with lines. They describe straight lines in the coordinate plane through equations often written in the form of \(y = mx + b\). Here's why linear equations are vital:

• They provide a straightforward relationship between two variables, usually x and y in Cartesian planes.
• Predict behavior: Knowing the equation allows you to predict the y-value for any given x.
• Identify parallel lines: Lines with the same slope (m) but different y-intercepts (b) will never cross each other.

In our specific exercise, the linear equations \(y = -0.5x + 5\), \(y = -0.5x - 1\), and \(y = -0.5x\) clearly demonstrate how linear equations are formulated. They help show parallelism in a straightforward way, which is a crucial property in geometry and real-world applications like designing roads or railway tracks. Connecting algebra to graphical interpretations enhances understanding and reinforces the foundational concepts of how these equations represent reality.