Problem 39 - Precalculus Mathematics for Calculus

Question

(a) Sketch the line with slope \(\frac{3}{2}\) that passes through the point \((-2,1)\) (b) Find an equation for this line.

Step-by-Step Solution

Answer

Equation: \(y = \frac{3}{2}x + 4\). Plot point \((-2, 1)\) and draw line from slope.

Step 1: Understanding the Slope-Intercept Form

The slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. We will use this form to find the equation of the line.

Step 2: Identify Given Information

We are given the slope \(m = \frac{3}{2}\) and a point \((-2, 1)\) through which the line passes. This point will help us determine the y-intercept \(b\).

Step 3: Use the Point-Slope Formula

The point-slope form of the equation is: \(y - y_1 = m(x - x_1)\). Substitute the slope \(m = \frac{3}{2}\) and the point \((-2,1)\) into the formula.

Step 4: Substitute Values into Point-Slope Formula

Substitute \( m = \frac{3}{2}\), \(x_1 = -2\), and \(y_1 = 1\) into the point-slope form: \[ y - 1 = \frac{3}{2}(x + 2) \]

Step 5: Simplify to Slope-Intercept Form

Distribute the \(\frac{3}{2}\) into \(x + 2\): \[ y - 1 = \frac{3}{2}x + 3 \] Add 1 to both sides to isolate \(y\): \[ y = \frac{3}{2}x + 4 \]

Step 6: Sketch the Line

Start by plotting the point \((-2, 1)\) on the graph. Use the slope \(\frac{3}{2}\), which means for every 2 units you move to the right, you move 3 units up, to draw a straight line passing through the point \((-2, 1)\). The line extends in both directions.

Step 7: Verify the Line Passes Through the Given Point

Substitute \(x = -2\) into the equation \(y = \frac{3}{2}x + 4\) to ensure that \(y\) equals 1: \[ y = \frac{3}{2}(-2) + 4 = -3 + 4 = 1 \] The calculation confirms the point \((-2, 1)\) is on the line.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a popular way to write the equation of a straight line. This form is expressed as \( y = mx + b \), where:

• \( m \) is the slope of the line, representing how steep the line is.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

The slope-intercept form is very useful because it immediately gives you two important pieces of information about the line—the slope and the y-intercept. When you graph a line using this form, start by plotting the y-intercept on the y-axis and then use the slope to find another point on the line. For example, a slope of \( \frac{3}{2} \) means that for each 2 units you move to the right, go up 3 units.

Point-Slope Formula

The point-slope formula is ideal for finding the equation of a line when you know the slope and a specific point on the line. It is given by the equation \( y - y_1 = m(x - x_1) \). Here:

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

To use the point-slope form, simply substitute the slope and the coordinates of the given point into the formula. For example, if the slope \( m \) is \( \frac{3}{2} \) and the line passes through \( (-2, 1) \), the equation becomes \( y - 1 = \frac{3}{2}(x + 2) \). This form is particularly useful when writing an equation because it can then be easily converted to slope-intercept form for graphing or analysis.

Graphing Lines

Graphing lines is a visual way to understand the relationship between two variables in a linear equation. To graph a line using an equation:

• First, identify the y-intercept \( b \) and plot it on the y-axis.
• Next, use the slope \( m \) to determine another point on the graph. The slope \( m = \frac{3}{2} \) means go up 3 units for every 2 units you move right.
• Draw the line through these points, extending it in both directions.

This gives a complete visual representation of the line. You can also use a given point, like \((-2, 1)\), plot it on your graph, and check if your line passes through it to verify your work. This method can be both engaging and intuitive for students.

Y-Intercept

The y-intercept is where the line crosses the y-axis. In the equation \( y = mx + b \), the term \( b \) represents the y-intercept. This is the value of \( y \) when \( x \) is zero. Determining the y-intercept can be simple if you know the slope and another point on the line.
In the example problem, using the point (-2,1) and slope \( \frac{3}{2} \), the equation simplifies to \( y = \frac{3}{2}x + 4 \), showing the y-intercept is 4. This value is crucial as it provides the starting point for graphing the line. Once you have the y-intercept:

• Plot it on the y-axis.
• Use the slope to find subsequent points.

Understanding the y-intercept helps in easily sketching the graph of a line and understanding its behavior.