Problem 26 - Precalculus with Limits

Question

In Exercises 17-28, find the slope and \(y\)-intercept (if possible) of the equation of the line. Sketch the line. \( y + 4 = 0 \)

Step-by-Step Solution

Answer

The slope of the line \(y + 4 = 0\) is 0 and the \(y\)-intercept is -4.

Step 1: Find the slope

For a horizontal line, the slope is always 0 because there is no change in \(y\) as \(x\) changes. So, the slope of this line is 0.

Step 2: Find the y-intercept

The y-intercept is the point at which the line crosses the y-axis. In the given equation \(y + 4 = 0\), if we rearrange it to \(y = -4\), we can see that the line crosses the y-axis at -4. So, the y-intercept of this line is -4.

Step 3: Sketch the line

Once we have the slope and y-intercept, we can sketch the line. Because the line has slope 0, it's a horizontal line. Draw a horizontal line that passes through the y-axis at -4.

Key Concepts

Slope of a line

The slope of a line is a way to measure how steep a line is. It's like asking how much does the line go up or down for each step it moves to the right. The slope is calculated by finding the change in the vertical direction (often called rise) divided by the change in the horizontal direction (called run). In mathematics, this is often represented with the formula \( m = \frac{\Delta y}{\Delta x} \).
For horizontal lines, where there is no vertical change, the slope is always 0. This is because the rise is zero, no matter how far you go along the x-axis. Imagine a flat surface—you can walk any distance across it without going uphill or downhill. That's why, in our example with the equation \( y + 4 = 0 \), the slope is 0.
Understanding slope is crucial because it helps us predict how a line behaves over different intervals and comprehend how steep or flat that line is. It dictates the direction and angle of the line we see on a graph.

y-intercept

The y-intercept is simply the point where the line crosses the y-axis. It's the value of \( y \) when \( x = 0 \). This is important because it gives us a starting point to graph the line among other ways to express line equations.
In slope-intercept form, an equation is written as \( y = mx + b \), where \( b \) is the y-intercept. It tells us exactly where on the y-axis the line will intersect. In our exercise example, when the equation \( y + 4 = 0 \) is rearranged to \( y = -4 \), it becomes clear that -4 is the y-intercept. The line crosses the y-axis at the point (0, -4).
Knowing the y-intercept allows you to immediately graph that point on the y-axis, making it easier to sketch the full line by finding other points.

Graphing lines

Graphing lines involves plotting a line on a graph using its equation. To graph a line, you need two key pieces of information: the slope and the y-intercept. These two pieces form the basic blueprint for constructing the line.

• Start by plotting the y-intercept on the graph—this is your first point.
• Use the slope to find the second point. Since the slope tells you how much the line goes up or down as you move right, count that many spaces up or down from the y-intercept, and one space right (since slope is often a fraction, you use the run as part of these instructions).

For a horizontal line, like in our example \( y + 4 = 0 \), with a slope of 0, you simply draw a straight line horizontally across the graph through the point \( y = -4 \). No need to calculate any additional points as the line doesn't rise or fall.
Graphing lines helps visually understand the relationships described by linear equations. It enables you to see points of intersection, compare different lines, and generally develop a deeper insight into linear functions.