Problem 46 - Algebra 2

Question

Graph the line with the given equation. \(y=-2 x\)

Step-by-Step Solution

Answer

Start at (0,0) and use slope -2 to plot and draw the line.

Step 1: Understand the Equation

The equation given is in the slope-intercept form, which is \( y = mx + b \). In this form, \( m \) is the slope of the line, and \( b \) is the y-intercept. For the equation \( y = -2x \), the slope \( m \) is -2 and the y-intercept \( b \) is 0.

Step 2: Identify the Slope

The slope \( m = -2 \) indicates the ratio of the rise over the run. This means for every 1 unit you move horizontally to the right (positive direction on the x-axis), the line moves 2 units downwards vertically (negative direction on the y-axis).

Step 3: Locate the Y-Intercept

The y-intercept \( b = 0 \) means that the line crosses the y-axis at the origin point (0,0). This is where you should start graphing the line.

Step 4: Plotting the Line

Start at the y-intercept point (0,0). From this point, use the slope to determine the next point. Since the slope is -2, move 1 unit to the right along the x-axis, and then move 2 units downwards along the y-axis to mark the next point. Repeat this to confirm the direction of the line.

Step 5: Draw the Line

Using a ruler, draw a straight line through the plotted points. Extend the line across the graph in both directions, making sure it accurately represents the slope and y-intercept from the equation. Label the line with its equation \( y = -2x \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a straightforward way to express a line on a graph. It is written as \( y = mx + b \). This format makes it easy to identify both the slope and the y-intercept of the line.

• \( m \) represents the slope of the line. This tells us how steep the line is and the direction in which it slants.
• \( b \) stands for the y-intercept, the point where the line crosses the y-axis. It is crucial for determining where to begin graphing the line.

With the slope-intercept form, graphing becomes simple as it provides a clear starting point and direction. Understanding this concept is helpful when moving from the algebraic equation to the geometric representation on a graph.

Y-Intercept

The y-intercept of a line, denoted by \( b \) in the slope-intercept form, is where the line meets the y-axis. It is a critical component because it serves as a starting point for graphing the line.

• If \( b \) is positive, the y-intercept is above the origin.
• If \( b \) is negative, it falls below the origin.
• When \( b = 0 \), the line crosses the origin at point (0, 0).

In our example, the equation is \( y = -2x \) with \( b = 0 \). This means the line begins at the origin, making this concept crucial when you start to plot your points. You always start drawing your line from the y-intercept when graphing.

Slope

The slope of a line is an indicative value that dictates how the line ascends or descends on the graph. Represented by \( m \) in the slope-intercept form \( y = mx + b \), the slope is calculated as the change in y over the change in x (rise over run).

• A positive slope means the line rises as it moves from left to right.
• A negative slope, as in \( m = -2 \), signals that the line descends.
• The steeper the slope, the more dramatic the line's rise or fall.

For instance, with a slope of \( -2 \), for every unit you move right on the x-axis, the line moves two units down. This direction is critical for plotting your line correctly, ensuring an accurate representation of the equation on the graph.