Problem 20 - Calculus with Concepts in Calculus

Question

Find an equation of the line described. Then sketch the line. The line with slope \(-2\) and \(y\) intercept \(\frac{5}{2}\)

Step-by-Step Solution

Answer

The equation is \( y = -2x + \frac{5}{2} \).

Step 1: Understand the Equation of a Line

The equation of a line in slope-intercept form is given by \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.

Step 2: Identify the Given Values

From the problem, we are given that the slope \( m \) is \(-2\) and the y-intercept \( c \) is \( \frac{5}{2} \).

Step 3: Substitute the Values into the Equation

Substitute \( m = -2 \) and \( c = \frac{5}{2} \) into the general equation to get: \[ y = -2x + \frac{5}{2} \].

Step 4: Sketch the Line

To sketch the line, plot the y-intercept \( \left(0, \frac{5}{2}\right) \) on the graph. Then, use the slope \(-2\), which means "down 2 units for every 1 unit to the right," to find another point on the line. Connect these points to draw the line.

Key Concepts

Equation of a Line

An equation of a line is like a blueprint for drawing a line on the coordinate plane. It tells us exactly how the line behaves by using two important pieces of information: the slope and the y-intercept. The slope-intercept form is the most common way to write an equation of a line, expressed as \( y = mx + c \).

• Here, \( m \) stands for the slope.
• \( c \) represents the y-intercept.

This format provides a clear picture of the line's direction and its point of intersection with the y-axis. By knowing the equation of a line, you can easily plot it on a graph and predict its path.

Slope

The slope of a line measures its steepness and direction. In the equation \( y = mx + c \), the slope is represented by \( m \). It determines how much the line rises or falls as you move along the x-axis. An easy way to understand it is by using the "rise over run" concept.

• If the slope is positive, the line ascends as it moves to the right.
• If the slope is negative, like \(-2\) in our exercise, the line descends as it moves to the right.
• A zero slope indicates a perfectly horizontal line.

A greater absolute value of the slope means a steeper line, while a smaller absolute value results in a flatter line. In our exercise, a slope of \(-2\) means the line goes down 2 units for every 1 unit it moves to the right. Understanding the slope helps in accurately drawing and interpreting lines.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + c \), the y-intercept is represented by \( c \). It's where the line meets the y-axis when the x-coordinate is zero.

• In our exercise, the y-intercept is \( \frac{5}{2} \) or 2.5, which means the line crosses the y-axis at this point.
• The y-intercept provides an anchor point, making it easier to draw or visualize the starting point of the line on a graph.
• It reveals the height of the line's starting point relative to the origin.

Plotting the y-intercept is the first step in sketching the line, followed by using the slope to determine its direction. This concept is crucial for graphing linear equations and understanding their behavior.