Problem 21 - 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 (-1,5) and is perpendicular to the line whose equation is \(x=6\).

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y=5\).

Step 1: Determine the slope

Since the line is perpendicular to \(x=6\), which is a vertical line, our line is a horizontal line. The slope of any horizontal line is 0, so \(m=0\).

Step 2: Substitute the point into the slope-intercept form

The line passes through the point (-1,5). Knowing that \(y = mx + b\) and \(m = 0\), we find the equation of the line by substituting the coordinates of the point and the slope into the equation: \(5 = 0*(-1) + b\).

Step 3: Solve for the y-intercept \(b\)

Solve the equation for \(b\), which would give \(b = 5\) as 0 times any number equals 0.

Key Concepts

Linear Function

A linear function is one of the most fundamental concepts in algebra and appears frequently in various mathematical applications. It is defined as a function that has a constant rate of change, which is graphically represented by a straight line. The general form of a linear function is expressed as
\( y = mx + b \).

Here, \( m \) denotes the slope, which measures the steepness of the line, indicating how much the \( y \)-value changes for each unit increase in the \( x \)-value. The \( b \) represents the y-intercept, which is the point where the line crosses the y-axis. This value tells us where the line hits the y-axis when \( x = 0 \). In the context of the original exercise, the function \( f \) being a linear function helps us to easily determine its equation. Since a horizontal line has a slope of 0, the linear function in the problem simplifies to \( y = b \), where \( b \) is the y-intercept. As the text continues, it's important to note that knowledge of the slope and the y-intercept enables students to graph the linear function and understand its behavior.

Perpendicular Lines

Perpendicular lines are a pair of lines that intersect at a right angle (90 degrees). In a coordinate plane, the relationship between the slopes of two perpendicular lines is particularly important. If one line has a slope of \( m \), then the slope of the line perpendicular to it will be \( -\frac{1}{m} \), which is the negative reciprocal of \( m \).

However, there is a special case when one of the lines is vertical (parallel to the y-axis) and has an undefined slope. A vertical line is described by an equation of the form \( x = c \), where \( c \) is a constant. When a line is perpendicular to a vertical line, it is a horizontal line, which has a slope of 0; its equation takes the simple form \( y = k \), where \( k \) is a constant, showing that \( y \) is the same no matter what \( x \) is. Using the original exercise as an example, the line given by \( x = 6 \) is vertical. Therefore, a line perpendicular to it would be horizontal and, consistent with the definition, has a slope of 0.

Y-intercept

The y-intercept is a significant characteristic of a linear function, as it provides a starting point for drawing the line on a coordinate plane. It is the y-coordinate of the point where the line crosses the y-axis. In other words, it's the value of \( y \) when \( x = 0 \). The slope-intercept form of a linear equation, \( y = mx + b \), readily gives the y-intercept as \( b \).

This value is crucial in graphing as well as understanding the function's behavior. From the exercise, the horizontal line has a y-intercept that remains constant across all values of \( x \), since the slope is 0. Therefore, the line described by the equation \( y = b \) reflects this fixed y-intercept. For the particular line in the exercise, after determining the slope to be 0, we simply used the given point (-1,5) to solve for the y-intercept (\( b \)), resulting in the y-intercept being 5.