Problem 20 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

After following the steps, we can graph the linear equation \(y=-\frac{1}{2} x+2\) using the points (-3, 3.5), (-2, 3), (-1, 2.5), (0, 2), (1, 1.5), (2, 1), (3, 0.5).

Step 1: Identify the slope and y-intercept

The equation given is in the slope-intercept form, with a slope of -1/2 and a y-intercept of 2. Hence, the graph will be a line with this slope and intercept.

Step 2: Substitute the given x-values

Now, substitute the x-values -3, -2, -1, 0, 1, 2, 3 in the given equation to find the corresponding y-values. For example, by substituting \(x = -3\) into the equation \(y = -\frac{1}{2}x + 2\), generate the pair (-3, 3.5)

Step 3: Repeat and get all pairs

Repeat the process for all other x-values to get the corresponding y-values. Afterwards, obtain all ordered pairs such as: (-3, 3.5), (-2, 3), (-1, 2.5), (0, 2), (1, 1.5), (2, 1), (3, 0.5).

Step 4: Plot the ordered pairs on a graph

Using the ordered pairs obtained from steps 2 and 3, plot these points on a coordinate plane. After marking the points, draw a straight line passing through them to represent the linear equation.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a mathematical equation of a line written as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis. This equation makes it simple to graph a linear equation by providing a starting point and a direction for drawing the line.

To graph a line using the slope-intercept form, you would first locate the y-intercept on the y-axis. In the exercise provided, the y-intercept is 2, so you would put a point on the y-axis at \(y = 2\). Next, using the slope, which in this case is \( -\frac{1}{2} \), you move from the y-intercept. The negative slope indicates that as \(x\) increases, \(y\) will decrease. For every 2 units you move to the right (the increase in \(x\)), you would move down (the decrease in \(y\)) by 1 unit to place your next point.

Understanding how to work with the slope-intercept form is fundamental for graphing linear equations efficiently.

Y-Intercept

The y-intercept is a fundamental characteristic of linear equations in the slope-intercept form. It is defined as the point on the graph where the line crosses the y-axis, which corresponds to \(x = 0\). The value of the y-intercept is represented by \(b\) in the slope-intercept equation \(y = mx + b\).

In the given exercise, the y-intercept is 2. This means that when \(x = 0\), the value of \(y\) is 2. You can double-check by substituting \(x = 0\) into the equation \(y = -\frac{1}{2}x + 2\), which indeed gives us \(y = 2\). For graphing purposes, the y-intercept is the starting point where you will begin plotting your line. It is worth noting that if the y-intercept is positive, the point lies above the origin, and if it is negative, the point lies below the origin.

Ordered Pairs

Ordered pairs are a pair of numbers used to locate points on a coordinate plane. They are written as \( (x, y) \), with \(x\) representing the horizontal position, and \(y\) representing the vertical position. In the context of graphing linear equations, finding ordered pairs involves substituting different \(x\)-values into the equation to solve for corresponding \(y\)-values.

For the exercise in question, a range of \(x\)-values were given (\-3, -2, -1, 0, 1, 2, 3). By substituting each of these values into the equation \(y = -\frac{1}{2}x + 2\), we find the corresponding \(y\)-values to create ordered pairs. These ordered pairs represent specific points on the graph that, when connected, form the line that represents the equation. Once you plot these points on your coordinate plane and connect them with a line, the visual representation of your linear equation is complete.