Problem 23 - Precalculus Essentials

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 line in slope-intercept form that passes through (-6,4) and is perpendicular to the given line is \(y = -1/2x + 1\).

Step 1: Determine the slope of the given line

The slope of a line can be found using two points on it by using the formula \((y2 - y1) / (x2 - x1)\). In this case, the points correspond to the intercepts, which are (2,0) for x-intercept, and (0,-4) for y-intercept. So, the slope (let's call this m1) of the given line is \((-4 - 0) / (0 - 2) = 2.

Step 2: Find the slope of the perpendicular line

The perpendicular slope, m2, is the negative reciprocal of the original slope, m1. Hence, m2 is equal to -1/m1 which is -1/2.

Step 3: Determine the equation of the line

The equation of a line in slope-intercept form is \(y = mx + c\) where m is the slope and c is the y-intercept. We know the slope, m2 is -1/2 and the line passes through the point (-6,4). We can substitute these values into \[y = mx + c\] to solve for c. Thus, \(4 = -1/2*(-6) + c\]. Solving this equation for c gives us c = 1.

Step 4: Write the equation of the line

Finally, substitute the slope and y-intercept values into the equation \(y = mx + c\). So, the equation of the line is \(y = -1/2x + 1\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a method of writing a linear equation so that the slope and y-intercept are immediately visible. This form is written as
\( y = mx + c \), where

• \( m \) is the slope of the line,
• \( c \) is the y-intercept, where the line crosses the y-axis.

For instance, to write an equation for a line that goes through a point and has a given slope, you can plug the slope into the \( m \) position and use the coordinates of the point to solve for \( c \).

Applying the Slope-Intercept Form

Given a point, like \( (-6, 4) \), and a slope of \( -1/2 \), you can find the y-intercept by substituting these values into the equation and solving for \( c \). Using the point to solve \( 4 = -1/2*(-6) + c \), you determine \( c = 1 \). Thus, the equation of the line is \( y = -1/2x + 1 \).

Perpendicular Lines

Perpendicular lines intersect each other at a 90-degree angle, and have slopes that are negative reciprocals of each other. In other words, if the slope of one line is \( a \), the perpendicular line will have a slope of \( -1/a \), assuming \( a \) is not zero. This is vital for finding the equation of a line that is perpendicular to another.

Identifying Perpendicular Slopes

If you have a line with the slope of 2, a perpendicular line will have a slope of \( -1/2 \). Doubling the negative reciprocal when working with perpendicular lines is a common pitfall to avoid. If the original slope is negative, the perpendicular slope will be positive, and vice versa.
By understanding how perpendicular slopes relate, you can accurately determine the orientation of lines in a coordinate plane and solve geometry and algebra problems involving perpendicular lines.

Slope of a Line

The slope of a line measures how steep the line is and the direction it slants. Calculated as the 'rise over the run', it's the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

• If the line moves upwards from left to right, the slope is positive,
• If it moves downwards, the slope is negative,
• If the line is horizontal, the slope is 0,
• And if it is vertical, the slope is undefined because you would be dividing by zero.

Calculating Slope

For example, take the line with an x-intercept of 2 and a y-intercept of -4. You can use these points to find the slope \( m \) using the formula:
\( m = \frac{(y2 - y1)}{(x2 - x1)} \). For our intercepts (2,0) and (0,-4), the slope is \( \frac{(-4 - 0)}{(0 - 2)} = 2 \). Comparing different lines' slopes can tell you how lines relate to one another, like if they're parallel (same slope) or perpendicular (negative reciprocals).