Problem 63 - Algebra 1

Question

Write the equation in slope-intercept form. Then graph the equation. $$y=-2$$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -2\), the slope is 0 and the y-intercept is -2. The graph is a horizontal line passing through the y-axis at -2.

Step 1: Write in Slope-Intercept Form

The equation \(y = -2\) is already in slope-intercept form. So, our \(m\) (slope) equals 0 and \(b\) (y-intercept) equals -2.

Step 2: Plot the Y-intercept

The y-intercept is the point where the line crosses the y-axis. Since \(b\) equals -2, the y-intercept is at the point (0, -2).

Step 3: Draw the Line

Because the slope of the line \(m\) is 0, this line is a horizontal line. From the y-intercept (0, -2), draw a horizontal line through this point.

Key Concepts

Graphing Equations

Graphing equations might seem daunting at first, but with the right approach, it becomes a clearer task. Essentially, when we graph an equation, we're visualizing a relationship between two variables, typically using the x-axis for one and the y-axis for the other. The most common form of an equation that we graph, especially when dealing with linear equations, is the slope-intercept form.
This form is written as:
\[y = mx + b\]

• "\(m\)" represents the slope of the line. This tells us how steep the line is and in which direction it leans.
• "\(b\)" indicates the y-intercept, the point where the line crosses the y-axis.

Once your equation is in this form, plotting it involves finding the y-intercept and then using the slope to determine another point on the line.
For horizontal or vertical lines, the graphing technique is even simpler since one of the axis values remains constant throughout. Whether complex or straightforward, graphing an equation allows us to see the connection between variables visually.

Horizontal Line

A horizontal line is one of the simplest graphs to plot. Unlike lines with slopes, where you have to consider upward or downward movements, a horizontal line remains perfectly flat. For equations like \(y = -2\), where the slope \(m\) is zero:

• The equation is already simplified as it shows the constant y-value across the graph.
• Since the slope is zero, no matter the x-value, the y-value remains constant.

In practical terms, to draw a horizontal line, start by marking the y-intercept (in our case, \(-2\) on the y-axis) and draw a straight line running left and right. It's important to remember that horizontal lines have zero slope, as there’s no incline; they purely reflect where the line sits vertically in relation to the x-axis.

Y-Intercept

The y-intercept of a line is the specific point where the line crosses the y-axis. In the slope-intercept form equation, \(y = mx + b\), the y-intercept is represented by "\(b\)."

• For the equation \(y = -2\), the y-intercept is \(-2\).
• This means on a graph, the line crosses the y-axis at the point (0, -2).

Knowing this point is crucial because it acts as the starting point for drawing the line on a graph.
Once the y-intercept is plotted, if the slope is given, one can determine the line's trajectory by following the slope.
However, if you're dealing with a horizontal line, as in this case, simply draw a straight line through the y-intercept across the graph, since the line remains at a consistent y-value.