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Chapter 1: Problem 20

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

Short Answer

Expert verified
The graph of the exercise represents a straight line with a negative slope of -1/2 and y-intercept of 2. The line decreases by 1/2 for each unit change in x.

Step by step solution

01

Identify the slope and y-intercept

From the equation \( y = -\frac{1}{2}x + 2 \), The slope, \( m \), is -1/2 and the y-intercept, \( b \), is 2. This means the line crosses the y-axis at y = 2, and for each move of 1 unit to the right in x, y will undergo 1/2 unit downwards.
02

Plot the y-intercept

On the graph, mark the point (0, 2). This is where the line crosses the y-axis.
03

Use the slope to plot more points

Use the slope of -1/2 to plot more points. Since the slope is negative, the line will go down as it goes to the right. From the y-intercept at (0,2), move 1 unit to the right and 1/2 unit downwards to get the point (1, 1.5). Similarly, you can move 1 unit to the left and 1/2 unit upwards to get the point (-1, 2.5). Repeat for all values of x provided.
04

Draw the line

Once you have all the points, draw a straight line that passes through all these points. This is your graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
Understanding the slope-intercept form is crucial for graphing linear equations. It is expressed as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) represents the y-intercept, which is where the line crosses the y-axis.

Take the equation \( y = -\frac{1}{2}x + 2 \) from our exercise as an example. The coefficient of \( x \), which is \( -\frac{1}{2} \), indicates the slope. The constant term, \( 2 \), indicates the y-intercept. This format allows for easy visualization of the line by first plotting the y-intercept and then using the slope to determine the direction and steepness of the line as it moves away from the y-intercept on the graph.
Plotting Points
Plotting points is the next step after identifying the slope and y-intercept. To plot points on a graph, you'll need a pair of values, one for the x-coordinate and one for the y-coordinate. For our exercise, with the given values of \( x \), we substitute them into the equation \( y = -\frac{1}{2}x + 2 \) to find the corresponding values of \( y \).

With each pair of \( (x, y) \), we make a dot on the graph at the intersection of those coordinates. It's important to be precise when plotting points because accurate placement leads to an accurate graph of the line.
Slope of a Line
The slope of a line is indicative of its rise over run, or how much the line goes up or down for a given horizontal distance. In the equation \( y = mx + b \), \( m \) is the slope. If it's positive, the line slants upwards to the right; if negative, as in our exercise with \( m = -\frac{1}{2} \), the line slants downwards as we move to the right.

To use the slope, start at the y-intercept, then move according to the slope's rise and run. For our negative slope, we move 1 unit down for every 2 units we move to the right. The slope is essential in defining the angle and direction of the line.
Y-intercept
The y-intercept, represented as \( b \) in the slope-intercept form, is the point where the line crosses the y-axis. It's the value of \( y \) when \( x = 0 \). In our equation \( y = -\frac{1}{2}x + 2 \), the y-intercept is \( 2 \). It is the starting point for plotting the linear equation on a graph.

When graphing, you always plot the y-intercept first. This point serves as an anchor from which you'll apply the slope to plot additional points. The y-intercept is a key landmark as it gives a physical location on the graph where the line will be positioned.

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Most popular questions from this chapter

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