Problem 24 - 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 (-5,6) and is perpendicular to the line that has an \(x\) -intercept of 3 and a \(y\) -intercept of -9.

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = -1/3x + 13/3\).

Step 1: Find the slope of the given line

The slope of a line is defined as the change in \(y\) divided by the change in \(x\) between two points on the line. Given that the line intercepts the x-axis at (3,0) and the y-axis at (0,-9), the slope \(m_1\) of that line can be calculated as \(m_1 = (y_2 - y_1)/(x_2 - x_1) = (-9 - 0)/(0 - 3) = 3 \).

Step 2: Determine the slope of the desired line

Since the desired line is perpendicular to the given, the slope \(m_2\) of the desired line is the negative reciprocal of \(m_1\). This can be calculated as \(m_2 = - 1/m_1 = - 1/3 \).

Step 3: Use the point-slope form to create the equation

Since we know the line passes through the point (-5,6), we can write the point slope form of the line, i.e., \(y - y_1 = m(x - x_1)\) where \((x_1, y_1)\) is the point on the line. Substituting the point (-5,6) and the slope \(-1/3\), we get \(y - 6 = -1/3 (x -(-5)) \).

Step 4: Convert to the slope-intercept form

Solve the equation from step 3 for \(y\), which gives us the slope-intercept form: \(y = mx + b\). Doing so, we find \(y = -1/3 x - 5/3 + 6 = -1/3 x + 13/3 \).

Key Concepts

Linear Function

A linear function is one of the simplest types of functions and is represented graphically by a straight line. The equation for a linear function is typically given in the format of a line or plane. The basic formula is

• Standard Form: \(Ax + By = C\)
• Slope-Intercept Form: \(y = mx + b\)

The slope-intercept form is particularly useful for quickly identifying the slope of the line \(m\) and its y-intercept \(b\).
This form makes it easy to graph the equation, as you can start at the y-intercept \(b\) and apply the slope \(m\) to find other points on the line.

Perpendicular Lines

Perpendicular lines intersect each other at a right angle, which is 90 degrees. This relationship between the lines affects their slopes.
In algebra, when two lines are perpendicular, the slopes of these lines have a special relationship: they are negative reciprocals of each other.
If the slope of one line is \(m_1\), the other line, perpendicular to it, will have a slope \(m_2\) such that:

• \(m_2 = -\frac{1}{m_1}\)
• This means if one line goes upwards \(+\), the other will go downwards \(-\) and vice versa

Understanding this property is crucial when determining the equation of a line that is perpendicular to another given line.

Negative Reciprocal

The concept of negative reciprocal is fundamental when dealing with perpendicular lines. A reciprocal of a number is

• fulfilling the property: if \(a = \frac{1}{b}\), then \(b = \frac{1}{a}\)

To find the negative reciprocal of a number or a slope, simply flip the numerator and the denominator of the fraction form of that number, and then change the sign.
For example, if the slope \(m_1 = 3\), the negative reciprocal, denoted as \(m_2\), will be \(m_2 = -\frac{1}{3}\).
This process is essential for identifying the slope of a line perpendicular to the one you're initially given.

Point-Slope Form

The point-slope form of a linear equation is used when you know the slope of a line and a point on the line. This form is beneficial in constructing the equation quickly and accurately.
The formula is:

• \(y - y_1 = m(x - x_1)\)

Where:

• \((x_1, y_1)\) is a point on the line
• \(m\) is the slope of the line

Using the point-slope form helps you connect all these parts together to form a linear equation. In the exercise, knowing that the slope is \(-1/3\) and the line passes through the point \((-5, 6)\), you can plug these values into the formula to find the equation of the line.
Afterwards, you can convert it into the slope-intercept form, which is often more convenient for graphing and interpreting.