Problem 47 - Algebra 1: Concepts and Skills

Question

Write the equation in slope-intercept form. Then graph the equation. $$ y=-4 $$

Step-by-Step Solution

Answer

The slope is 0 and the y-intercept is -4. The graph will be a horizontal line crossing the y-axis at the point (0, -4).

Step 1: Identify the slope and y-intercept

In the equation \(y = -4\), there is no slope (m) represented. This indicates that the slope is 0, making the line horizontal. The y-intercept (b) is represented by -4, meaning the line crosses the y-axis at the point (0, -4).

Step 2: Graph the y-intercept

Begin by drawing a coordinate plane. Plot the y-intercept at (0, -4). This is the point where our line will cross the y-axis.

Step 3: Graph the line

Since our slope is 0, we know the line is horizontal, only running along the y-axis. Remember, the slope gives us a rise/run. With a slope of 0, this indicates there is no rise, only run. So our line is a flat line, horizontally crossing the y-intercept point (0, -4). Simply draw a straight horizontal line running through the y-point -4 to finish the graph of the equation.

Key Concepts

Graphing Linear Equations

When graphing linear equations, it's important to understand the slope-intercept form of a line, which is given by the equation \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) is the y-intercept of the line.
In our example, the equation is \(y = -4\). This lacks an \(x\) term, emphasizing its unique nature. When you graph this line on the coordinate plane, you start by finding the y-intercept. The line crosses the y-axis at \(-4\).

• Identify the y-intercept: In this case, it's -4.
• Graph the y-intercept on the coordinate plane at \((0, -4)\).
• If the slope was present, you would use it to find more points by following "rise over run" logic, but here it's zero.

Once plotted, you can draw a straight line through this y-intercept, noting that for a horizontal line, the graphing step becomes simplified. You don't need to adjust for rise or run since the line remains consistently flat across the plane.

Horizontal Line Equation

A horizontal line is specifically represented by the equation \(y = c\), where \(c\) is any constant value. In equations like \(y = -4\), the value -4 signifies that the line will always have the same y-coordinate regardless of the x-coordinate.
A few key points about horizontal lines:

• They have a slope of 0.
• The line runs parallel to the x-axis.
• The y-coordinate is constant across all points on the line.

Drawing horizontal lines is straightforward: you identify the constant (in this case, \(-4\)), and draw a line that never veers up or down from that y-level across the entire graph.

Y-Intercept

The y-intercept \(b\) in the slope-intercept form \(y = mx + b\) tells us where the line will cross the y-axis. It is the point on the line when the x-value is zero.
In the equation \(y = -4\), the y-intercept is \(-4\), which means the line will cross the y-axis at the point \((0, -4)\). This provides immediate information about the position of the line on the graph.
Understanding the y-intercept is essential as:

• It provides the starting point for graphing the line.
• It's a primary characteristic of the line's behavior.

Particularly in horizontal lines, the y-intercept effectively describes the entire line since there is no slope to alter its direction.