Problem 19 - College Algebra

Question

Graph each equation .Let $x=-3,-2,-1,0,1,2, and 3. $$ y=-\frac{1}{2} x $$

Step-by-Step Solution

Answer

To graph the equation \(y=-\frac{1}{2} x \), first substitute the x-values (-3, -2, -1, 0, 1, 2, 3) into the equation to get the corresponding y-values. These will be: (1.5, 1, 0.5, 0, -0.5, -1, -1.5) respectively. Then, plot these points and draw a line through them. The line will have a negative slope.

Step 1: Substitute x-values in the equation

First, we have to substitute the given values of x in our equation \( y = -\frac{1}{2} \times x \) in order to find the corresponding y-values. For x ranging from -3 to 3, we will get: When \( x = -3, y = 1.5 \); \(x = -2, y = 1 \); \(x = -1, y = 0.5 \); \(x = 0, y = 0 \); \(x = 1, y = -0.5 \); \(x = 2, y = -1 \); \(x = 3, y = -1.5\).

Step 2: Plotting the points

Now, we'll plot these points (-3, 1.5), (-2, 1), (-1, 0.5), (0, 0), (1, -0.5), (2, -1), and (3, -1.5) on a graph.

Step 3: Draw the line

With all points plotted, the next step is to draw a straight line through these points that will represent the equation \(y=-\frac{1}{2} x \). This line is a downward line representing negative slope. It starts from the upper left corner and moves to the bottom right corner of the graph.

Key Concepts

Graphing Equations

Graphing equations is a way to visually represent how variables interact within an equation. For instance, when you graph a linear equation, it often creates a straight line. This is due to the constant relationship between the variables. Let's consider the equation given in the exercise:

• Equation: \( y = -\frac{1}{2} x \)

In this equation, \( y \) changes by \(-\frac{1}{2}\) for each increment of 1 that \( x \) increases. To graph this equation, you follow a series of steps:

• Choose a set of values for \( x \) (in this case, values from -3 to 3).
• Calculate the corresponding \( y \) values using the equation.
• Plot these \( x, y \) pairs on a coordinate plane.
• Connect the dots to reveal how \( y \) behaves as \( x \) changes.

Graphing gives a clear visual understanding of the relationship and provides insights into the equation's behavior.

Negative Slope

The concept of slope in a linear equation describes the steepness and direction of the line. When we have a negative slope, it indicates that as \( x \) increases, \( y \) decreases. In our equation:

• Equation: \( y = -\frac{1}{2} x \)
• Slope: \(-\frac{1}{2}\).

The negative sign in the slope signifies a downward trend on the graph. This means the line will start high on the left and move lower as it progresses to the right. It's essential to understand that the negative slope effectively shows an inverse relationship between \( x \) and \( y \). As one variable goes up, the other comes down. Graphing a line with a negative slope can often provide a better comprehension of how inputs relate through a reduction in the dependent variable.

Plotting Points

Plotting points involves placing specific coordinates on a graph to visually represent data. This process is the foundation for drawing any graph, including lines derived from equations. In our exercise:

• The points calculated from \( x = -3 \) to \( x = 3 \) gave us ordered pairs like \((-3, 1.5)\) and \((3, -1.5)\).
• Each of these pairs indicates a position on the graph where the line passes through.

To plot a point, you find the \( x \) value on the horizontal axis and the corresponding \( y \) value on the vertical axis. Placing a dot where these two values meet establishes the point.

• Start from the center, move left or right for the \( x \) value.
• Then, move up or down for the \( y \) value.

Repeating this for each pair ensures that you lay the groundwork for the line. When correctly plotted, the points should all line up, confirming the accuracy of your graphing.