Problem 11 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$ $$ \text { Perpendicular to } y-\pi=\pi(x-1) \text { with a } y \text { -intercept of } 3 $$

Step-by-Step Solution

Answer

The equation of the line that is perpendicular to \(y-\pi=\pi(x-1)\) and that has a y-intercept of 3 is \(y = -\frac{1}{\pi}x + 3\)

Step 1: Find the slope of the given line.

Given the line equation \( y-\pi = \pi(x-1) \), we can simplify this to standard form, which is \(y = mx + c\), as follows: \(y = \pi x + (\pi - \pi)\). Thus, the slope of the given line is \(\pi\).

Step 2: Find the slope of the line that is perpendicular to the given line

The slope of the line perpendicular to the given line is the negative reciprocal of the slope of the given line. Hence, if the slope of the given line is \(\pi\), the slope (\(m'\)) of the line that is perpendicular to it is \(-1/\pi\).

Step 3: Use the y-intercept and the slope of the perpendicular line to form the equation of the new line

Now, the equation of the new line will be in the form \(y = m'x + b\), where \(m'\) is the slope of the new line and \(b\) is the given y-intercept (3 in this case). Substituting the values for \(m'\) and for the y-intercept we have: \(y = -\frac{1}{\pi}x + 3\)

Key Concepts

Slope of a Line

The slope of a line, often represented as "m" in the equation of a line, is a measure of its steepness. It tells us how vertical or how flat a line is.
In mathematical terms, the slope is defined as the change in the vertical direction (y-axis) divided by the change in the horizontal direction (x-axis) between two points on the line.
To find the slope from an equation like this one:

• Identify the equation's form. The most common form is the slope-intercept form: \[y = mx + c\]
• Here, "m" represents the slope of the line.

For example, in the equation: \[y = \pi x\], the slope "m" is \(\pi\). A higher slope means a steeper line, while a slope of zero means a flat line. Understanding slope is crucial for determining the direction and steepness of a line.

Perpendicular Lines

Perpendicular lines are fundamental in geometry and algebra.
They intersect at a right angle, specifically 90 degrees.
When studying linear equations, it's crucial to know how to identify them. The slopes of two perpendicular lines are negative reciprocals of each other.
What does this mean?

• If one line has a slope of \(m\), then the perpendicular line will have a slope of \(-1/m\).
• This relationship is based on the idea that the product of their slopes is -1.

Let's consider the initial line from the step-by-step solution:

• The given line has a slope of \(\pi\).
• The perpendicular line, therefore, has a slope of \(-1/\pi\).

This concept is used to find the equation of a line that is perpendicular to another, using their slope relation.

Y-Intercept

The y-intercept of a line is a key characteristic in the slope-intercept form. It is often denoted by "c" or "b" and represents the point where the line crosses the y-axis.
It's generally expressed as a single point: \((0, c)\) or \((0, b)\).
This point gives crucial information about the starting point of the line when the x-value is zero.

• In the equation \(y = mx + c\), "c" is the y-intercept.
• If \(c = 3\), the line crosses the y-axis at \((0, 3)\).

Understanding the y-intercept is important for graphing lines and determining where they intersect with the y-axis. In the given solution, the y-intercept of the perpendicular line was directly provided, \(b = 3\), making it straightforward to form its equation.