Problem 26 - Precalculus Essentials

Question

Write an equation in slope-intercept form of a linear function \(f\) whose graph satisfies the given conditions. The graph of \(f\) is perpendicular to the line whose equation is \(4 x-y-6=0\) and has the same \(y\) -intercept as this line.

Step-by-Step Solution

Answer

The linear function \(f\) which is perpendicular to the given line and shares its y-intercept can be represented by the equation \(y = -1/4x - 6\).

Step 1: Identify the slope and y-intercept of the given line

Transform the equation of the given line into slope-intercept form, \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept. This can be done by rearranging the equation \(4x-y-6=0\) to the form \(y = 4x -6\). Thus, the slope is 4 and the y-intercept is -6.

Step 2: Determine the slope of the desired function

The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Since the slope of the given line is 4, the slope of the function \(f\) is the negative reciprocal of 4, or \(-1/4\).

Step 3: Write the equation of the desired function using its slope and y-intercept

The function \(f\) shares its y-intercept with the given line, so its y-intercept is also -6. Using the slope-intercept form, we can write the equation of the function \(f\) as \(y = -1/4x -6\).

Key Concepts

Linear Equations

Linear equations are a fundamental concept in mathematics and are widely used to describe straight lines in a coordinate plane. A linear equation takes the form of \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. To understand how a line behaves, especially in how we describe lines in algebra, we often convert this general form into the slope-intercept form.

Slope-intercept form is written as \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. This form makes it easier to quickly identify how steep the line is and where it crosses the y-axis.

• The slope \(m\) reflects the angle of the line: a larger slope means a steeper line.
• The y-intercept \(b\) is the point where the line intersects the y-axis.

This form is particularly useful for graphing and understanding the relationship between variables in a linear equation.

Perpendicular Lines

Perpendicular lines are an important concept in geometry, crucial for forming right angles. Two lines are perpendicular if they intersect at a 90-degree angle. When working with linear equations in slope-intercept form, the slopes of perpendicular lines have a unique relationship.

If you have a line with a slope \(m\), a line that is perpendicular to it will have a slope that is the negative reciprocal of \(m\). This simply means flipping the fraction and changing the sign.

• For example, if the slope of one line is \(4\), the slope of a line perpendicular to it will be \(-1/4\).

Understanding this relationship allows you to quickly determine the slope of a line that will be perpendicular to another, which is a common task in problems involving geometry and coordinate systems.

Y-Intercept

The y-intercept is a key part of any linear equation because it tells you where the line crosses the y-axis on a graph. In the slope-intercept form \(y = mx + b\), \(b\) represents the y-intercept.

This value is crucial for sketching the graph of the equation since it provides a starting point. Knowing the y-intercept allows for ease in visualizing the graph's orientation along the y-axis.

• When given a specific y-intercept, you can place a dot on the graph where the line will cross the y-axis.
• This feature is shared in all lines with the same constant \(b\) in their equation, meaning multiple lines can share the same y-intercept like in our exercise.

This concept not only aids in graphing but also in understanding how different lines can interact or share characteristics in a coordinate plane.