Problem 84 - Precalculus: Functions and Graphs

Question

Many calculators have viewing sereens that are wider than they are high. The approximate ratio of the height to the width is often \(2: 3 .\) Let the actual height of the calculator screen along the \(y\) -axis be 2 units, the actual width of the calculator screen along the \(x\) -axis be 3 units, and Xscl \(=\mathbf{Y s c}=1 .\) since the line \(y=x\) must pass through the point \((1,1),\) the actual slope \(m_{A}\) of this line on the calcuIator screen is given by $$m_{A}=\frac{\text { actual distance between tick marks on } y \text { -axis }}{\text { actual distance between tick marks on } x \text { -axis }}$$ Using this information, graph \(y=x\) in the given viewing rectangle and predict the actual angle \(\boldsymbol{\theta}\) that the graph makes with the \(x\) -axis on the viewing screen. \([0,6]\) by \([0,2]\)

Step-by-Step Solution

Answer

The line appears with a slope \(m_A = \frac{3}{2}\), resulting in an angle of approximately \(56.31^\circ\) with the x-axis on the screen.

Step 1: Understanding the Problem

We need to graph the line \(y=x\) on a screen where the height and width have a ratio of \(2:3\). We also need to determine the actual slope \(m_A\) and predict the angle \(\theta\) the graph makes with the x-axis based on the screen dimensions.

Step 2: Determine the Aspect Ratio Controls

The line \(y=x\) typically has a slope of 1, but the screen distorts this due to its 2:3 aspect ratio. We will calculate this distortion by finding \(m_A\), the actual slope of the line within the viewing rectangle.

Step 3: Calculate the Actual Slope \(m_A\)

The formula for \(m_A\) is \(\frac{\text{actual distance between tick marks on } y\text{-axis}}{\text{actual distance between tick marks on } x\text{-axis}}\). Here, each unit on the y-axis equates to 1 actual unit, and each unit on the x-axis is \(\frac{2}{3}\) actual units. Thus, \(m_A = \frac{1}{\frac{2}{3}} = \frac{3}{2}\).

Step 4: Predict the angle \(\theta\)

The slope \(m_A = \frac{3}{2}\) is the tangent of the angle \(\theta\). Use the inverse tangent to find \(\theta\): \[\theta = \tan^{-1}\left(\frac{3}{2}\right) \approx 56.31^\circ\].

Step 5: Graph the Line \(y = x\)

Within the viewing rectangle \([0,6]\) by \([0,2]\), plot points that satisfy \(y=x\), taking into account that due to the screen's scaling, the line will appear to have a steeper slope, reflecting the increased angle.

Key Concepts

Slope Calculation

When working with graphs, remembering that the slope of a line is crucial. The slope tells us how steep a line is. It is calculated as the change in y-values divided by the change in x-values between two points on a line. On a standard graph without distortions, the line \( y = x \) has a slope of 1, meaning it rises one unit for every unit it runs to the right.

However, when the actual screen dimensions come into play, the apparent slope changes due to distortion. This distortion is captured by the actual slope, denoted as \( m_A \). In this exercise, since the units are scaled differently along the axes, the slope \( m_A \) becomes \( \frac{3}{2} \). This calculation assumes the y-axis runs at a regular scale, but the x-axis units are compressed to two-thirds their actual width, altering the line's steepness visually.

Inverse Tangent

The tangent of an angle relates to the slope of a line in the context of trigonometry. In simple terms, it tells us the ratio of the opposite side to the adjacent side of a right triangle formed by the line.

To find the angle that corresponds to a given slope, we use the inverse tangent function, often notated as \( \tan^{-1} \). For the line \( y = x \) on the distorted screen with \( m_A = \frac{3}{2} \), the angle \( \theta \) can be calculated using this function.

• The inverse tangent of \( \frac{3}{2} \) is \( \theta \).
• Mathematically, \( \theta = \tan^{-1}\left(\frac{3}{2}\right) \).
• This results in an angle of approximately \( 56.31^\circ \).

Understanding this lets us see how changes in aspect ratio affect the visual angle of lines on the display.

Graphing Techniques

To graph a line effectively, especially on a screen with specific aspect ratios, you need to consider how the display settings alter its representation. Here's how you can go about it:

• Start by identifying the key points that satisfy the line equation, such as points on the line \( y = x \) within the rectangle \([0, 6]\) by \([0, 2]\).
• Plot these points accurately considering the distortion induced by the screen's aspect ratio.
• Connect these points forming a line that naturally would have been drawn at a 45-degree angle, but now appears steeper due to the compressed x-axis.

This cautious approach helps ensure that the graph reflects both the mathematical model and the graphical reality dictated by the screen dimensions.

Screen Distortion

Screen distortion is an important concept, particularly when translating graphs from theory onto visual mediums like calculator displays. The distortion occurs because the screen's physical dimensions do not match the perfectly square unit grid that mathematical graphs typically assume.

In this case, the height-to-width aspect ratio of 2:3 means the screen is wider than it is tall. This difference scales the graph horizontally more than vertically, affecting all graphics displayed.

• The x-axis compresses by one-third its original scale, making all slopes appear steeper.
• The line \( y = x \), having an apparent slope \( m_A \), reflects this change.

Understanding screen distortion helps us comprehend why graphs might not appear as we mathematically expect, especially for functions or lines typically having a certain angle or slope relative to axes.