Question

a) Graph the line \(y=\frac{3}{4} x,\) where \(x>0\) Mark an angle \(\theta\) that represents the angle formed by the line and the positive \(x\) -axis. Plot a point with integral coordinates on the line \(y=\frac{3}{4} x\) b) Use these coordinates to determine \(\tan \theta\) c) Compare the equation of the line with your results in part b). Make a conjecture based on your findings.

Short answer

The line's slope is \(\tan\theta\). For \(y=\frac{3}{4}x\), \(\theta\) is such that \(\tan\theta = \frac{3}{4}\).

Step-by-step solution

Graph the line

To graph the line, we start by using the equation provided: \[y = \frac{3}{4}x \]Since we're only interested in the portion of the graph where \(x > 0\), we plot several positive \(x\) values and find the corresponding \(y\) values. For example, if \(x = 4\), then \(y = \frac{3}{4} * 4 = 3\). Plot the points (4, 3), (8, 6), etc.

Mark the angle \(\theta\)

Mark the angle \(\theta\) between the graphed line and the positive \(x\)-axis. This is done where the line crosses the \(x\)-axis.

Plot a point with integral coordinates

From Step 1's plotted points, choose a point with integral coordinates such as (4, 3). Ensure this point is clearly marked on the graph.

Determine \(\tan\theta\)

Determine \(\tan\theta\) using the coordinates of the plotted point. Recall that \(\tan\theta = \frac{\text{opposite}}{\text{adjacent}}\). Here, \(\tan\theta = \frac{3}{4}\) because the point (4, 3) corresponds to (\(x, y\)) = (4, 3).

Compare with the line equation

Compare the result from Step 4 with the given line equation. Notice that \(\frac{y}{x} = \frac{3}{4} = \tan\theta\), which implies the slope \(m\) of the line \(y = \frac{3}{4}x\) is the tangent of the angle \(\theta\).

Make a conjecture

Based on the comparison in Step 5, conjecture that for any line \(y = mx\), the slope \(m\) is equal to \(\tan\theta\), where \(\theta\) is the angle formed by the line with the positive \(x\)-axis.

Key concepts

Slope

In the context of linear equations, the slope is a critical measure. It indicates how steep the line is and is represented by the letter 'm' in the equation of a line in the form of \(y = mx + b\). In the given exercise, the equation is \(y = \frac{3}{4} x\), and the slope is \(\frac{3}{4}\).
The slope tells us that for every 4 units we move horizontally to the right (increase in \(x\)), we move 3 units vertically up (increase in \(y\)). This is why when we plot points like (4, 3) and (8, 6), they lie on the same straight line. Understanding slope helps in graphing the equation and analyzing the relationship between \(x\) and \(y\).

Tangent of Angle

The tangent of an angle \(\theta\) in trigonometry is defined as the ratio of the opposite side to the adjacent side in a right triangle. Mathematically, it is expressed as \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\). In the exercise, \(\theta\) is the angle formed by the line with the positive \(x\)-axis.
When we use the point (4, 3) to calculate \(\tan \theta\), we find that \(\tan \theta = \frac{3}{4}\). This shows the direct relationship between the slope of the line and the tangent of the angle it forms with the \(x\)-axis. This relationship helps us understand how angles are related to the linear equations in coordinate geometry.

Coordinate Geometry

Coordinate geometry involves plotting and analyzing points, lines, and shapes using coordinate points. In the exercise, we plot the line \(y = \frac{3}{4}x\) by choosing values for \(x > 0\) and calculating corresponding \(y\) values.
We plotted points like (4, 3) and (8, 6), which lie on the line. These points have integral coordinates that are easy to plot on the graph. By relating these points and the line they form, we gain insights into the relationship between algebraic equations and their graphical representations.
This graphical interpretation makes abstract algebraic concepts more concrete and visually understandable.

Trigonometry

Trigonometry provides tools to study the relationships involving angles and distances. In this exercise, we use trigonometry to determine \(\tan \theta\) for the angle formed by the line with the positive \(x\)-axis.
We know that for any right triangle, \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\). Using the coordinates of the point (4, 3), we see that it fits into the formula as \(\frac{3}{4}\). This not only helps in calculating the tangent but also illustrates a deeper connection between algebraic expressions of lines and trigonometric ratios.
Understanding this concept is key to solving many geometric and real-world problems involving angles and distances.