Problem 63 - Brief Applied Calculus

Question

Use your graphing calculator to graph the following four equations simultaneously on the window \([-10,10]\) by \([-10,10]:\) $$ \begin{array}{l} y_{1}=2 x+6 \\ y_{2}=2 x+2 \\ y_{3}=2 x-2 \\ y_{4}=2 x-6 \end{array} $$ a. What do the lines have in common and how do they differ? b. Write the equation of another line with the same slope that lies 2 units below the lowest line. Then check your answer by graphing it with the others.

Step-by-Step Solution

Answer

The lines have the same slope (2) but different y-intercepts. The new line is \(y = 2x - 8\).

Step 1: Enter the Equations into the Calculator

First, turn on your graphing calculator. Enter the equations into the calculator by accessing the 'Y=' graphing menu. Input each line as follows: - For \(y_1\), enter \(2x+6\)- For \(y_2\), enter \(2x+2\)- For \(y_3\), enter \(2x-2\)- For \(y_4\), enter \(2x-6\).

Step 2: Set the Graphing Window

Set the window to - Xmin: -10 - Xmax: 10 - Ymin: -10 - Ymax: 10. Ensure that you have a suitable scale that allows you to see each line clearly.

Step 3: Graph the Equations

After setting the window, hit the 'Graph' button. You should now see four straight lines displayed on your calculator screen. Observe the relationship between these lines.

Step 4: Analyze the Graph

All four lines should appear parallel as they share the same slope of 2 but differ in their y-intercepts. - The common feature is the slope (steepness), which is consistently 2 for all lines, indicating they rise 2 units for every 1 unit they move horizontally. - The difference is in their vertical placement, determined by their y-intercepts: 6, 2, -2, and -6 respectively.

Step 5: Find Equation of a New Line

To find an equation for a line parallel to these but 2 units below the lowest line, start with the lowest line, which is \(y = 2x - 6\). Subtract 2 from the y-intercept to slide the line 2 units down: New line equation: \(y = 2x - 8\).

Step 6: Verify by Graphing the New Line

Return to the 'Y=' menu on the calculator, and enter \(y_5 = 2x - 8\). Rerun the graph and check that this new line is parallel to the others and lies 2 units below \(y_4\).

Key Concepts

Parallel lines

Parallel lines are lines that never meet, no matter how far they extend in either direction. They run alongside each other, maintaining a constant distance apart. This means they have the same slope when graphed on a coordinate plane. Understanding parallel lines is pivotal in graphing linear equations because:

• Lines with identical slopes are parallel. For example, in the original exercise, all four lines share a slope of 2. This common slope ensures they are parallel to each other.
• While the slope (steepness of the line) is the same, the lines have different y-intercepts, which shift them up or down vertically in the graph.
• In practical terms, if you need a line parallel to another, just keep the slope the same, and adjust the y-intercept to shift the line vertically.

Graphically, if two lines on a graph have the same angle of incline and maintain this positioning, they will never cross, making them parallel. Recognizing this helps in predicting behavior of linear systems and organizing variables efficiently.

Slope-intercept form

The slope-intercept form of a line is an equation of the form \(y = mx + b\). Here, \(m\) represents the slope of the line while \(b\) denotes the y-intercept. This formula is a staple in graphing linear equations because of its simplicity and readability:

• Slope \((m)\): It indicates how steep the line is. A larger slope means a steeper incline, while a smaller slope results in a flatter line. In this exercise, each line has a slope of 2, indicating a consistent steepness that moves two units up for every unit moved right.
• Y-intercept \((b)\): It's where the line crosses the y-axis. Changing this value shifts the line up or down. For instance, lines from the exercise have y-intercepts of 6, 2, -2, and -6, placing them at different starting points along the y-axis.

To alter a line's vertical position without changing its steepness, adjust the y-intercept, as done when finding a line parallel two units below another. Comprehending slope-intercept form is essential for quickly determining and graphing lines efficiently.

Graphing calculator

A graphing calculator is a powerful tool that aids in visualizing mathematical concepts, especially in graphing linear equations. Here’s how they streamline the process:

• Inputting Equations: Easily enter equations using the 'Y=' function. For the exercise, we input lines like \(y_1 = 2x + 6\) directly to visualize them promptly.
• Setting Up a Window: Define a specific graphing window, such as [-10, 10] by [-10, 10], to frame the view appropriately, ensuring all lines fit within and can be observed clearly.
• Rendering the Graph: By pressing 'Graph', these devices plot entries instantaneously, demonstrating the relationships (like parallelism) between equations visually.

Using a graphing calculator not only confirms theoretical work but also provides a tangible view of how varying elements, like y-intercepts, affect positioning of lines. They are invaluable for students in understanding data, functions, and real-time alterations.