Problem 126 - A Graphical Approach to Precalculus with Limits

Question

In a square window of your calculator that gives a good picture of the first quadrant, graph the line \(y=\sqrt{3} x\) with \(x \geq 0 .\) Then, trace to any point on the line. See the figure. What we see is the terminal side of a standard position angle in quadrant I. Store the values of \(\mathrm{X}\) and \(\mathrm{Y}\) in convenient memory locations, and call them \(\mathrm{X}_{1}\) and \(\mathrm{Y}_{1}\). Look at the equation of the line you graphed, and make a conjecture: The _______ of a line passing through the origin is equal to the _______ of the angle it forms with the positive \(x\) -axis.

Step-by-Step Solution

Answer

The slope of a line passing through the origin is equal to the tangent of the angle it forms with the positive x-axis.

Step 1: Understand the Line Equation

The equation of the line is given by \( y = \sqrt{3} x \). This is a linear equation representing a line passing through the origin (0,0) with a slope of \( \sqrt{3} \). Since \( x \geq 0 \), we will only consider the part of the line in the first quadrant.

Step 2: Graph the Line Equation

On your graphing calculator, enter the equation \( y = \sqrt{3} x \) and set the viewing window to capture the first quadrant appropriately. This will typically be done by setting both the x-axis and y-axis to range from 0 to positive values. Graph the line and observe its trajectory.

Step 3: Identify a Point on the Line

Once the line is graphed, use the trace function on your calculator to move to any point on the line. This will give you specific \( x \) and \( y \) values that lie on the line. Let's label these \( X_1 \) and \( Y_1 \). For example, if you trace to the point where \( x = 1 \), \( y \) will be \( \sqrt{3} \times 1 = \sqrt{3} \).

Step 4: Store the Coordinates

Store the coordinates of the traced point in memory. This makes it easy to recall \( X_1 \) (the x-coordinate) and \( Y_1 \) (the y-coordinate) for further analysis. These coordinates should satisfy \( Y_1 = \sqrt{3} \times X_1 \).

Step 5: Analyze the Line's Slope

The slope \( m \) of the line is \( \sqrt{3} \). This represents the change in \( y \) per unit change in \( x \). In trigonometry, the slope of a line passing through the origin is equal to \( \tan(\theta) \), where \( \theta \) is the angle the line makes with the positive x-axis.

Step 6: Make a Conjecture

Based on the analysis, we conjecture that the slope of the line (\( \sqrt{3} \)) is equal to \( \tan(\theta) \), where \( \theta \) is the angle it forms with the positive x-axis. Therefore, the slope of the line is equal to the tangent of the angle it makes with the x-axis. Therefore, the conjecture is: The slope of a line passing through the origin is equal to the tangent of the angle it forms with the positive x-axis.

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in precalculus that helps us understand the steepness or incline of a line. When working with the slope, think of it as the 'rise over run'. This means how much the line rises vertically for a unit change horizontally.
For the line given by the equation \( y = \sqrt{3} x \), the slope is \( \sqrt{3} \). This value tells us that the line rises \( \sqrt{3} \) units for every 1 unit it runs horizontally. Slopes can be interpreted as:

• Positive: the line ascends as it moves to the right.
• Negative: the line descends as it moves to the right.
• Zero: the line is horizontal.
• Undefined: the line is vertical.

Since our line passes through the origin and heads to the first quadrant, it exhibits a positive slope.

Graphing Calculator

Using a graphing calculator can significantly simplify the process of understanding mathematical concepts visually. These calculators allow you to graph equations, locate points of interest, and enhance comprehension of geometric relationships.
To graph the equation \( y = \sqrt{3} x \) on a graphing calculator:

• Enter the equation in the graphing function of the calculator.
• Set the viewing window to show the first quadrant where \( x \geq 0 \).
• Utilize the trace function to follow along the line, which helps to understand how the slope operates across various points.

By practicing these steps, you can see the trajectory of the line clearly. This also leads to a deeper understanding of how the line interacts with the axes.

Standard Position Angle

The concept of a standard position angle is one where the angle's vertex is at the origin of the coordinate plane, and its initial side lies along the positive x-axis.
For the line given by \( y = \sqrt{3} x \):

• The terminal side of the angle stretches from the origin, through the quadrant, adhering to the line's path.
• Since our line lies in the first quadrant, the angle is between 0° and 90°.
• The tangent of this angle can be derived from the line's slope.

This viewpoint allows us to see geometric transformations and consider relationships in trigonometry, especially how the angle aligns with axes.

Tangent Function

In trigonometry, the tangent function, \( \tan(\theta) \), relates to how steep an angle is when compared to the x-axis. Essentially, it compares the opposite side to the adjacent side in a right triangle. For any given line passing through the origin, the slope is directly linked to this function.
For the equation \( y = \sqrt{3} x \):

• The slope \( \sqrt{3} \) is equal to \( \tan(\theta) \).
• This means the angle \( \theta \), where \( \theta = \tan^{-1}(\sqrt{3}) \), determines how the line intersects the x-axis.
• This specification gives the angle at which the line inclines relative to a horizontal baseline.

Recognizing this connection is crucial for applying trigonometric functions to solve real-world problems involving angles and their respective slopes.