Problem 38 - Precalculus

Question

(a) The lines \(y=4 x\) and \(y=-0.25 x\) are perpendicular because their slopes are negative reciprocals. Use a graphing utility to graph these two lines in the standard viewing rectangle. Unless your graphing utility automatically shows true proportions, the lines will not appear to be perpendicular. (b) If necessary, modify the viewing rectangle in part (a) so that true proportions are used and the two lines indeed appear perpendicular.

Step-by-Step Solution

Answer

Adjust the graph to equal scaling on both axes to make the lines appear perpendicular.

Step 1: Understand the Slopes of the Lines

The lines given are \(y = 4x\) and \(y = -0.25x\). The slope of the first line is \(m_1 = 4\) and the slope of the second line is \(m_2 = -0.25\). When two lines are perpendicular, the product of their slopes should be \(-1\). Calculating: \(m_1 \times m_2 = 4 \times (-0.25) = -1\). This confirms they are perpendicular.

Step 2: Graph the Lines in the Standard Viewing Rectangle

Use a graphing tool such as Desmos or a graphing calculator. Plot the lines \(y = 4x\) and \(y = -0.25x\) using the standard viewing rectangle, usually with x and y ranging from -10 to 10. Observe how the lines appear in this default setting.

Step 3: Analyze the Appearance of the Lines

Even though the lines are mathematically perpendicular, they might not appear perpendicular due to the scaling of the axes if they are not equal, i.e., the units on the x-axis are not the same as the units on the y-axis in the standard view.

Step 4: Modify the Viewing Rectangle for True Proportions

To make the lines appear truly perpendicular, adjust the graph such that both the x and y axes use the same scale. This will often mean using a square graphing window, ensuring equal units along both axes. Set both axes to have the same range, such as -10 to 10 on both axes.

Step 5: Verify the Perpendicular Appearance

Check the appearance of the lines in this modified graph. Now that the axes are equally scaled, the lines should visually appear to intersect at a 90-degree angle, confirming they're perpendicular.

Key Concepts

Graphing Utility

When tackling graphing tasks, a graphing utility is an indispensable tool. It refers to any software or device, like Desmos or TI-84 calculators, that helps visualize mathematical functions by plotting them on a coordinate plane. This powerful tool allows you to instantly see the graph of an equation, making it much easier to understand the relationship between variables.

To effectively use a graphing utility, follow these steps:

• Input the equations you need to graph. In our case, these are the lines given as equations, like \(y=4x\) and \(y=-0.25x\).
• Select a viewing window that represents the range of values for x and y you are interested in. Most utilities have default settings, but you might need to adjust these for accurate depictions.
• Use the tool's features to analyze the intersection or behavior of the lines. It might have options for adjusting line appearance or changing viewing angles.

Taking full advantage of a graphing utility can help illuminate concepts such as slope and the appearance of perpendicular lines in different settings.

Negative Reciprocal

The term 'negative reciprocal' plays a key role in understanding perpendicular lines. Mathematically, if two lines have slopes that are negative reciprocals of each other, they are perpendicular.

To find the negative reciprocal of a number:

• First, take the reciprocal of the number, which is simply flipping the number's fraction. For instance, the reciprocal of 4 is \(\frac{1}{4}\).
• Then, apply the negative sign. So, the negative reciprocal of 4 is \(-\frac{1}{4}\).

In the case of our lines, the slopes are 4 and -0.25. Calculating the product \(4 \times -0.25 = -1\) confirms they're negative reciprocals. This connection enables us to conclude that lines with slopes which multiply to \(-1\) are perfectly perpendicular.

Slope

The slope of a line is a fundamental concept in graphing. It gives us a measure of how steep a line is, defined typically as 'rise over run.' This means how much the line goes up (or down) vertically for a given movement horizontally.

For a linear equation in the form \( y = mx + b \), the slope is represented by the coefficient \( m \). For example:

• In \( y = 4x \), the slope \( m \) is 4, indicating the line rises 4 units for every 1 unit it runs right.
• In \( y = -0.25x \), the slope \( m \) is -0.25, indicating the line falls 0.25 units for every unit it runs right.

The slope not only determines the line's steepness but also aids in checking perpendicularity between lines, as perpendicular lines have slopes that are negative reciprocals of each other.

Viewing Rectangle

The viewing rectangle is crucial in graphing as it sets the range of values the graph will display on both axes. This range determines what is visible on your graphing utility screen and can greatly affect the perception of the graph.

In a standard graphing utility, the viewing rectangle typically ranges from -10 to 10 on both axes. However, this default may not always show accurate geometrical relationships.
For perpendicular lines to appear truly perpendicular, especially when their slopes are negative reciprocals, the viewing rectangle must be set to equal scales on both axes.

• This means ensuring that one unit on the x-axis equals one unit on the y-axis.
• Adjusting the window to have equal dimensions, such as \(-10\) to \(10\) on both axes, ensures accurate portrayal of angles.

By modifying the viewing rectangle to a square window, you can visually confirm perpendicular relationships between lines accurately.