Problem 20 - Applied Calculus

Question

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. $$ y=-3 $$

Step-by-Step Solution

Answer

Slope \( m = 0 \), y-intercept \((0, -3)\). The graph is a horizontal line at \( y = -3 \).

Step 1: Identify the Equation Type

The given equation is \( y = -3 \). This is a linear equation where \( y \) is constant and does not depend on \( x \).

Step 2: Determine the Slope

Since \( y \) does not change with \( x \) (as there is no \( x \) term in the equation), the slope \( m \) is 0. This means the line is horizontal.

Step 3: Find the y-intercept

The y-intercept \((0, b)\) is the point where the graph crosses the y-axis. Since \( y = -3 \) for all values of \( x \), the y-intercept is \((0, -3)\).

Step 4: Sketch the Graph

Draw a horizontal line across the coordinate plane at \( y = -3 \). All points on this line have y-coordinate of -3, indicating that the line does not rise or fall as \( x \) changes.

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial when working with linear equations. Slope is a measure of how steep a line is. Think of it as how much the line "goes up" for each step you take horizontally. Mathematically, this is often expressed as the change in the vertical direction (rise) over the change in the horizontal direction (run).

This concept is usually associated with a variable called "m". For instance, in an equation like \( y = mx + b \), "m" represents the slope. A positive slope indicates the line rises as you move from left to right, while a negative slope tells you the line falls. In the given exercise, the equation is \( y = -3 \). Here, there is no \( x \) term, meaning the slope \( m \) is 0.

• Zero slope: The line is horizontal and does not rise or fall.
• Positive slope: The line increases from left to right.
• Negative slope: The line decreases from left to right.

When the slope is zero, as in this exercise, the line remains perfectly flat across the graph, indicating constancy in its direction.

Y-Intercept

The y-intercept is the point where your line touches or crosses the y-axis. This special point provides valuable information about the equation's behavior, especially when graphing it. Essentially, it tells us the value of \( y \) when \( x \) is zero.

In the equation \( y = mx + b \), "b" is the y-intercept. It's the constant term. For the exercise you have, the equation is \( y = -3 \). This means that for every \( x \), \( y \) is consistently \(-3\). Thus, the y-intercept is the point \((0, -3)\).

• Y-intercept of \( (0, -3) \) signals that the line crosses the y-axis at this point.
• It's a fixed point on the graph where the line intersects the y-axis.
• Provides a starting point when sketching the graph.

Even if the equation lacks other terms, the y-intercept gives a solid ground for starting your graphing work.

Graphing Linear Equations

Graphing linear equations is a visual way to represent how an equation looks on a coordinate plane. By plotting the y-intercept first and then using the slope, you can sketch the entire line step-by-step.

For the equation \( y = -3 \):

• Identify the slope and y-intercept first. The slope here is \( 0 \), and the y-intercept is \( (0, -3) \).
• Since the slope is \( 0 \), the line is horizontal. This means you draw a straight line parallel to the x-axis, passing through \( y = -3 \).
• Each point on the line will have a y-value of \(-3\), regardless of the x-value.

Drawing the graph helps visualize the equation, reinforcing your understanding of how these elements interact. Each point on this horizontal line mirrors the lack of variation in the y-coordinate, thanks to the slope of zero. Graphs are an invaluable tool, offering a clear snapshot of equations at a glance.