Problem 20 - College Algebra

Question

Graph each equation in Exercises \(13-28 .\) Let \(x=-3,-2,-1\) \(0,1,2,\) and 3. $$y=-\frac{1}{2} x+2$$

Step-by-Step Solution

Answer

The graph will be a straight descending line which passes through the points calculated by substituting the given \(x\) values (-3, -2, -1, 0, 1, 2, and 3) into the equation. After calculating the corresponding \(y\)-values and plotting the points, a line that best fits these points would give the graph of the equation \(y=-\frac{1}{2} x+2\).

Step 1: Calculate Y-Values

Start by substituting the given \(x\) values into the equation. This means replacing \(x\) with -3, -2, -1, 0, 1, 2, and 3 respectively, each time calculating the resultant \(y\) value.

Step 2: Plot the Points on the Graph

After determining the \(y\)-values, we will have a series of points, each represented as (x, y). Plot all of these points on the graph.

Step 3: Draw the Line

Once all the points are plotted on the graph, draw a line that passes through them. The plotted points should align in a straight line, representing the equation. The line should slope downward from left to right, in accordance with the negative coefficient of \(x\) in the equation.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward way to express the equation of a straight line. This form is given by the equation: \(y = mx + b\), where:

• \(m\) represents the slope of the line.
• \(b\) is the y-intercept, or where the line crosses the y-axis.

In our problem, the equation \(y = -\frac{1}{2}x + 2\) is already in slope-intercept form.
This makes it easy to identify the slope and the y-intercept directly from the equation. Here, the slope (\(m\)) is \(-\frac{1}{2}\), which indicates that the line falls as you move from left to right.
The y-intercept (\(b\)) is 2, meaning the line crosses the y-axis at the point (0, 2).
Understanding this form helps you quickly graph lines and predict their behavior without first calculating individual points.

Coordinate Plane

Graphing linear equations requires understanding the coordinate plane. The coordinate plane is a two-dimensional space marked by a horizontal axis (x-axis) and a vertical axis (y-axis).
These axes intersect at a point called the origin, labeled as (0, 0). The plane is used to represent pairs of numbers visually.

• The first number in the pair is the x-coordinate, which tells how far to move horizontally.
• The second number is the y-coordinate, which tells how far to move vertically.

When graphing the equation \(y = -\frac{1}{2}x + 2\), you'll plot points based on the x and y values generated from the equation.
Each point is represented as a coordinate pair (x, y).
Understanding how to navigate the coordinate plane is key in plotting these points accurately.

Plotting Points

Plotting points is an essential skill for graphing linear equations. Once you have calculated the y-values for given x-values, such as x = -3, -2, -1, etc., you create points like (-3, y), (-2, y), (-1, y), and so on. Here's how to plot these points:

• Start by locating the x-value on the x-axis.
• Then, move vertically to the y-value. If y is positive, move up; if y is negative, move down.
• Place a dot where this vertical line intersects the x-coordinate.

Repeat this process for each x-value to get a series of points.
In the given exercise, the coordinates calculated from the equation \(y = -\frac{1}{2}x + 2\) will produce points through which a line can be drawn.
Plotting accurate points ensures that your graph reflects the correct slope and position of the line in the coordinate plane.