Problem 21 - College Algebra

Question

Write an equation in slope-intercept form of a linear function \(f\) whose graph satisfies the given conditions. The graph of \(f\) passes through \((-6,4)\) and is perpendicular to the line that has an \(x\) -intercept of 2 and a \(y\) -intercept of \(-4\)

Step-by-Step Solution

Answer

The equation of the function in slope-intercept form is \( y = - \frac{1}{2}x + 1 \).

Step 1: Find the slope of the given line.

The equation of the given line can be written using the slope-intercept form (y = mx + b) where m is the slope and b is the y-intercept. From the given points, which are the x and y intercepts, we can determine that the slope of the line (m) is change in y (rise) over change in x (run) which gives \[ m = \frac{-4 - 0}{0 - 2} = 2 \]. So, the slope of the line is 2.

Step 2: Determine the slope of our line.

Two lines are perpendicular if and only if the product of their slopes is -1. Therefore, the slope of the line that is perpendicular to a line with a slope 2 is the negative reciprocal of 2. Which is \[ m = - \frac{1}{2} \].

Step 3: Derive the equation of the line.

Now that the slope of our required line is known, its equation can be derived using the given point (-6,4) and the slope-intercept form, which is y = mx + b. First, plug in the known values: 4 = -1/2 * -6 + b -> 4 = 3 + b. Solve this equation for b to find the y-intercept, which results in b = 1. Therefore, the equation of the line is \( y = - \frac{1}{2}x + 1 \).

Key Concepts

Perpendicular Lines

When we say two lines are perpendicular, we mean they intersect at a right angle (90 degrees). The slopes of perpendicular lines have a special relationship: their product is -1. This is because changing the direction exactly by 90 degrees flips the slope in a mathematically unique way.
Understanding this concept is crucial when dealing with linear equations involving perpendicular lines.
Here’s a simple way to remember:

• If one line has a slope of \( m \), then the perpendicular line's slope will be \( -\frac{1}{m} \).
• In our example, the given line has a slope of 2. So, the perpendicular line's slope is \( -\frac{1}{2} \).

This flipping of the slope can help you quickly find equations of lines that are perpendicular.

x-intercept

The x-intercept of a line is the point where the line crosses the x-axis. At this point, the value of \( y \) is zero. Finding the x-intercept is straightforward: set \( y = 0 \) and solve for \( x \).
In the problems you're solving, knowing the x-intercept helps to form the equation of the line.
For example:

• If you know a line crosses the x-axis at \( x = 2 \), then the point is \( (2, 0) \).
• This kind of information, combined with the y-intercept, allows you to determine the slope of a line by considering how much \( y \) changes as \( x \) changes.

This basic understanding is essential when writing linear equations, especially in slope-intercept form.

y-intercept

The y-intercept is the point where a line crosses the y-axis. Here, the value of \( x \) is zero. Finding the y-intercept is crucial as it is often used directly in the slope-intercept form of a line, \( y = mx + b \), where \( b \) represents the y-intercept.
Understanding the y-intercept helps in creating and interpreting linear equations:

• For instance, if a line crosses the y-axis at \( y = -4 \), then the point is \( (0, -4) \).
• The y-intercept tells you where the line will start when plotted on a graph, providing a baseline for your line's slope to extend from.

In both visualization and equation solving, the y-intercept plays a key role in detailing the line's path.

Linear Equations

Linear equations form the foundation of algebra and are vital in mathematics. These equations represent lines and are usually written in the slope-intercept form: \( y = mx + b \).
Here are some key points to grasp:

• \( m \) is the slope, which indicates how steep the line is.
• \( b \) is the y-intercept, showing where the line crosses the y-axis.
• This equation form makes it easy to plot and understand the behavior of lines on a graph.

By manipulating the values of \( m \) and \( b \), one can determine how the line behaves and where it is positioned in relation to the axes.
Whether the line is horizontal, vertical, or at an angle, the linear equation provides a simple way to describe its nature mathematically.