Problem 25 - 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 \(3 x-2 y-4=0\) and has the same \(y\) -intercept as this line.

Step-by-Step Solution

Answer

The equation of the line perpendicular to the given line and with the same y-intercept is \(y = 2/3x + 2\).

Step 1: Find the slope of the given line

To find the slope of the given line, first arrange the equation into slope-intercept form \(y= mx+c\) where \(m\) is the slope. Here, the equation is \(3x - 2y - 4 = 0\). So the slope \(m_1\) of the given line is \(-3/2\).

Step 2: Calculate the slope of the perpendicular line

The slope of any line perpendicular to a given line is the negative reciprocal of the slope of this given line. So, for our line, the slope \(m_2\) is the negative reciprocal of \(-3/2\), which equals \(2/3\).

Step 3: Find the y-intercept of the given line

The y-intercept of the given line is calculated by setting \(x = 0\) in the equation. If we do so, we have -2y - 4 = 0, thus y-intercept \(c\) is 2.

Step 4: Write the equation of the perpendicular line

Now that we have the slope \(m_2 = 2/3\) and the intercept \(c = 2\), we can substitute these values into the formula \(y = mx+c\). Which gives us the equation of the perpendicular line \(y = 2/3x+2\).

Key Concepts

Understanding Perpendicular Lines

Perpendicular lines are important in geometry and algebra. When two lines intersect at a right angle, they are perpendicular to each other. This relationship affects their slopes. If you have a line with a slope of \( m_1 \), a line perpendicular to it will have a slope of \( m_2 \) which is the negative reciprocal of \( m_1 \).
For example:

• If \( m_1 = -\frac{3}{2} \), the slope \( m_2 \) would be \( \frac{2}{3} \). This was used in our exercise to find the equation of the line that is perpendicular to the given line.

Understanding how to calculate the perpendicular slope helps in finding new equations, especially in problems where right angles or orthogonal relationships are involved.

Identifying the Y-Intercept

The y-intercept is where a line crosses the y-axis. In the equation of a line \( y = mx + c \), the \( c \) represents the y-intercept.
This tells us the value of \( y \) when \( x = 0 \).
To find it in the given line equation \( 3x - 2y - 4 = 0 \), set \( x \) to zero and solve for \( y \). This revealed the y-intercept as \( 2 \).

• Y-intercepts are crucial for graph sketching.
• They provide the starting point of a linear equation.

With this knowledge, you can easily draw graphs or solve problems by understanding where the line begins on the y-axis.

Basics of Linear Functions

Linear functions describe a straight line in the coordinate plane and are usually expressed in the slope-intercept form \( y = mx + c \).
Here:

• \( m \) is the slope, indicating the steepness or direction of the line.
• \( c \) is the y-intercept, showing where the line crosses the y-axis.

Linear functions are simple yet powerful in modeling relationships between variables. They have:

• A constant rate of change.
• A predictable pattern, which makes them easy to work with.

Mastery of linear functions enables better understanding of more complex mathematical concepts and practical applications like predicting trends and outcomes.