Problem 19 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph of the equation \(y=-\frac{1}{2}x\) will be a straight line passing through the points (-3, 1.5), (-2, 1), (-1, 0.5), (0, 0), (1, -0.5), (2, -1), and (3, -1.5).

Step 1: Substitute the x-values into the equation

Start by substituting the given x-values into the equation to calculate corresponding y-values. These calculations are as follows: \(y=-\frac{1}{2}(-3) = 1.5\), \(y=-\frac{1}{2}(-2) = 1\), \(y=-\frac{1}{2}(-1) = 0.5\), \(y=-\frac{1}{2}0 = 0\), \(y=-\frac{1}{2}(1) = -0.5\), \(y=-\frac{1}{2}(2) = -1\) and \(y=-\frac{1}{2}(3) = -1.5\)

Step 2: Plot the points on the graph

Now, these x and y value pairs are points on the line represented by the equation \(y=-\frac{1}{2}x\). Plot these points on a graph. The points are (-3, 1.5), (-2, 1), (-1, 0.5), (0, 0), (1, -0.5), (2, -1), and (3, -1.5)

Step 3: Draw the line

Once all the points have been plotted, the final step is to connect these points to form a straight line. The connection of these points gives the graphical representation of the equation \(y=-\frac{1}{2}x\).

Key Concepts

Plotting Points

Understanding how to plot points on a coordinate plane is essential for graphing linear equations. The coordinate plane consists of two perpendicular number lines that intersect at the origin, labeled as (0,0). The horizontal number line is called the x-axis, while the vertical number line is called the y-axis. Each point on the plane is determined by an ordered pair \( (x, y) \), where \( x \) represents the horizontal position, and \( y \) the vertical position.

To plot a point, find the \( x \) value on the x-axis and the \( y \) value on the y-axis. Your point will be located where a vertical line from the x-value intersects a horizontal line from the y-value. For the equation \( y = -\frac{1}{2}x \) from the exercise, you'd start by substituting specific x-values to get the corresponding y-values. Once calculated, you can plot these pairs like (-3, 1.5), (-2, 1), etc., on the plane. It's like placing a dot on the canvas of the coordinate plane at the precise location defined by each \( (x, y) \) pair.

Linear Function Graph

A linear function graph is a straight line on the coordinate plane and is the graphical representation of a linear equation, such as \( y = mx + b \). In this form, \( m \) is the slope, and \( b \) the y-intercept, where the line crosses the y-axis.

Linear functions demonstrate a constant rate of change, which you can see as the slope of the line. When graphing \( y = -\frac{1}{2}x \) as given in our exercise, after plotting the points, you'll notice they all align to form a straight line, indicative of its linear nature. The slope here is -\frac{1}{2}, meaning for every unit we move to the right on the x-axis, the value of \( y \) decreases by half a unit. The y-intercept is at (0, 0) for this particular equation, which means the line crosses the origin.

Slope of a Line

The slope of a line is a measure of its steepness and the direction in which it tilts. It's an important concept when graphing linear equations because it defines the angle and direction of the line on the coordinate plane. The slope is typically represented by the letter \( m \) and is calculated by the formula \( m = \frac{\text{change in y}}{\text{change in x}} \) or \( m = \frac{\Delta y}{\Delta x} \).

For the line represented by the equation \( y = -\frac{1}{2}x \) from our exercise, the slope is -\frac{1}{2}. This negative slope indicates that the line slopes downward from left to right. To visualize this, consider any two points on the line—such as (-2, 1) and (2, -1)—and notice how for a 4 unit increase in \( x \) (from -2 to 2), \( y \) decreases by 2 units (from 1 to -1), which is consistent with the slope of -\frac{1}{2}. Understanding the slope helps in predicting other points on the line beyond those calculated and enhances your comprehension of linear relationships.