Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$ $$ \text { Passing through points }(\pi, 3) \text { and }(-\pi, 5) $$

Step-by-Step Solution

Answer

The equation of the line that passes through the points \((\pi, 3)\) and \((-\pi, 5)\) is \(y = -\frac{x}{\pi} + 4\)

Step 1: Calculate the slope

The slope \(m\) of the line through two points \((x1, y1)\) and \((x2, y2)\) is given by the formula \(m = \frac{{y2 - y1}}{{x2 - x1}}\). So, using the provided points \((\pi, 3)\) and \((- \pi, 5)\), the slope would be \(m = \frac{{5 - 3}}{{- \pi - \pi}} = \frac{{2}}{{- 2\pi}} = - \frac{1}{\pi}\)

Step 2: Use one point and the slope to find the equation of the line

We can use the point-slope form of the line which is \(y - y1 = m(x - x1)\). We substitute one of the points, say \((\pi, 3)\), and the value of the slope \(m = - \frac{1}{\pi}\) from Step 1, into the formula. Doing so will give us the equation of the line, \(y - 3 = -\frac{1}{\pi}(x - \pi)\)

Step 3: Simplify the equation

On expanding the equation from Step 2, we get \(y-3 = -\frac{x}{\pi} + 1\). Lastly, we rearrange this equation to find an equivalent, but more standard, linear equation. Adding \(3\) and \(\frac{x}{\pi}\) to both sides to isolate \(y\) gives us the final equation \(y = -\frac{x}{\pi} + 4\).

Key Concepts

Slope Calculation

The slope of a line is a measure of how steep the line is. It describes the change in the vertical direction (up or down) relative to the change in the horizontal direction (left or right).

To calculate the slope between two points, you use the formula:

• \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \)

Where \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of two points on the line.

In the given exercise, you have the points \((\pi, 3)\) and \((-\pi, 5)\).

Plug these values into the slope formula:
\[m = \frac{{5 - 3}}{{-\pi - \pi}} = \frac{2}{-2\pi} = -\frac{1}{\pi}\\]This negative value indicates the line is going downward as it moves from left to right. Understanding slope allows you to see the general direction and steepness of a line.

Point-Slope Form

The point-slope form is a versatile way to find the equation of a line when you know the slope and one point on the line. The point-slope equation is:

• \(y - y_1 = m(x - x_1)\)

Here, \((x_1, y_1)\) is the known point on the line and \(m\) is the slope.

In this problem, you have the slope \(m = -\frac{1}{\pi}\) from our previous calculation, and you can use either point, \((\pi, 3)\), in the equation:

\[y - 3 = -\frac{1}{\pi}(x - \pi)\\]This equation represents the line in its basic point-slope form and retains all the information about the line’s slope and one specific point.

This form is particularly helpful because you can easily rearrange it to find the line's equation in other forms, such as standard or slope-intercept form.

Two-Point Formula

The two-point formula is a method used to determine the equation of a line using two known points.

This approach integrates slope calculation and the point-slope form to directly find the linear equation. First, calculate the slope using the formula discussed in the first section, providing a slope of \(-\frac{1}{\pi}\).

Then, plug this slope and one of the points, say \((\pi, 3)\), into the point-slope form:

• \(y - 3 = -\frac{1}{\pi}(x - \pi)\)

Solve for \(y\) to simplify the equation further:
\[y - 3 = -\frac{x}{\pi} + 1\\]By rearranging, you get the linear equation:
\[y = -\frac{x}{\pi} + 4\\]This technique is neat because it blends concepts and shows the progression from knowing points to arriving at a line's full equation. Whether you're aiming for the slope-intercept form or any other type of line equation, the two-point formula serves as a strong foundation.