Problem 95 - College Algebra with Modeling and Visualization

Question

Graph the lines \(y=2 x\) and\(y=-\frac{1}{2} x\) in the standard viewing rectangle. (a) Do the lines appear to be perpendicular? (b) Graph the lines in the following viewing rectangles.i. \([-15,15,1]\) by \([-10,10,1]\) \ddoti. \([-10,10,1]\) by \([-3,3,1]\) in. \([-3,3,1]\) by \([-2,2,1]\) Do the lines appear to be perpendicular in any of these viewing rectangles? (c) Determine the viewing rectangles where perpendicular lines will appear perpendicular. (Answers may vary.)

Step-by-Step Solution

Answer

Yes, the lines are perpendicular; appearance varies by rectangle settings.

Step 1: Identify Line Equations

We have two lines given by the equations: \( y = 2x \) and \( y = -\frac{1}{2}x \). These lines have slopes of 2 and \(-\frac{1}{2}\), respectively.

Step 2: Determine Perpendicularity

Two lines are perpendicular if the product of their slopes is -1. Calculate the product of the slopes: \( 2 \times -\frac{1}{2} = -1 \). Since the product is -1, the lines are indeed perpendicular.

Step 3: Graph the Lines in the Standard Viewing Rectangle

Graph the lines \( y = 2x \) and \( y = -\frac{1}{2}x \) using a standard viewing rectangle. These lines will intersect at the origin and form a 90-degree angle, appearing perpendicular.

Step 4: Graph in Viewing Rectangle [-15,15,1] by [-10,10,1]

When graphed within this rectangle, the lines still cross at the origin, maintaining their perpendicular orientation. They should appear perpendicular.

Step 5: Graph in Viewing Rectangle [-10,10,1] by [-3,3,1]

In this rectangle, the vertical stretch is more significant. These lines will still cross at the origin and could potentially appear perpendicular, but viewer perception may vary due to stretching.

Step 6: Graph in Viewing Rectangle [-3,3,1] by [-2,2,1]

This viewing rectangle is quite narrow on the y-axis, possibly distorting appearance. The lines cross at the origin, but they may not appear perpendicular visually due to the limited y-axis range.

Step 7: Identify Ideal Viewing Rectangles for Perpendicularity

Lines appear most naturally perpendicular when both axes have equal or proportional scales. Larger rectangles that encompass more distance on both axes can mitigate visual distortion.

Key Concepts

Perpendicular Lines

When we say two lines are perpendicular, we mean they intersect at a perfect right angle of 90 degrees. This special relationship between lines can be identified through the product of their slopes. If the slopes of two lines multiply together to give -1, the lines are perpendicular. For instance, consider the lines described by the equations: \( y = 2x \) and \( y = -\frac{1}{2}x \). The slopes are 2 and \(-\frac{1}{2}\), respectively. Calculating their product gives:

• \( 2 \times -\frac{1}{2} = -1 \)

As the result is -1, these lines are perpendicular. A neat way to visualize perpendicular lines is by sketching them so they intersect at the origin, forming neat cross-like shape.

Slopes of Lines

The slope of a line tells us how steep it is. It is a measure of how much the line goes up or down as we move from left to right. The standard formula for the slope (m) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]For the lines \( y = 2x \) and \( y = -\frac{1}{2}x \), the slopes are 2 and \(-\frac{1}{2}\), respectively:

• A positive slope, like 2, indicates the line rises as you move right.
• A negative slope, like \(-\frac{1}{2}\), indicates the line falls as you move right.

Understanding slopes helps us determine if lines are parallel (same slope) or perpendicular (product of slopes is -1). It sets the groundwork for perceiving the orientation of lines on a graph.

Viewing Rectangles

Viewing rectangles refer to the range we choose for the x- and y-axes when graphing functions. This range impacts how lines and graphs are displayed, either highlighting or obscuring important features. Typical viewing rectangle settings might look something like:

• \([-15, 15, 1]\) by \([-10, 10, 1]\)
• \([-10, 10, 1]\) by \([-3, 3, 1]\)
• \([-3, 3, 1]\) by \([-2, 2, 1]\)

These settings dictate the min and max values for each axis, and how much space we have to observe the graph's behavior. Using a standard rectangle often gives a balanced view of the graph, while customized rectangles can magnify certain aspects or fit the graph to a specific context. It's crucial when confirming concepts like perpendicularity, where the visual confirmation complements the mathematical proof.

Visual Distortion

Visual distortion can occur in graphing when the viewing rectangle skews how lines are perceived, especially when the axis scales are unequal. Even if lines are mathematically perpendicular, they might not appear so if the stretching or compressing of the viewing rectangle is significant. Consider a narrow rectangle like \([-3, 3, 1]\) by \([-2, 2, 1]\). Here, the y-axis range is pretty small compared to the x-axis range. Such disproportionate scaling affects how angles and intersections appear, making perpendicular lines seem anything but 90 degrees.

• Equal or proportional length scales on both axes help maintain line orientation.
• Larger viewing frames may reduce distortion effects, providing a clearer picture.

Understanding and managing visual distortion is key in graphing, ensuring what you see mirrors mathematical truth.