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Chapter 6: Problem 109

Graph: \(y=-\frac{1}{2} x+2 .\) (Section 3.4, Example 3)

Short Answer

Expert verified
The graph of the equation \(y=-\frac{1}{2} x + 2\) is a line that intercepts the y-axis at point (0,2) and has a slope of -1/2, which means for every 2 units move to the right, there is a 1 unit move down.

Step by step solution

01

Identify the slope and y-intercept

The given equation is in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. From the given equation \(y = -\frac{1}{2}x + 2\), we can identify that the slope \(m\) is -1/2 and the y-intercept \(c\) is 2.
02

Plot the y-intercept

From the earlier step, we identified that the y-intercept is 2. The y-intercept is the point where the line crosses the y-axis. This will be the first dot in the graph, at point (0, 2).
03

Use the slope to plot the second point

The slope is the ratio of the vertical change to the horizontal change, often denoted as 'rise over run'. Our slope is -1/2, so from the y-intercept, we go down 1 unit (rise of -1, as the slope is negative), and run 2 units to the right (run of 2). Plot this second point, which should be at (2, 1).
04

Draw the line

Now we have two points. Draw a line passing through these points, which will give us the graph of the linear equation.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most commonly used formats. It's written as \( y = mx + c \), where \( m \) represents the slope and \( c \) is the y-intercept. This form is incredibly helpful because it allows you to quickly identify both the slope and the y-intercept of the line without additional calculations. For students, mastering this form is essential because it simplifies the process of graphing and understanding linear equations. It tells you not only about the direction and steepness of the line but also where exactly it crosses the y-axis.
Slope Calculation
Understanding how to calculate the slope is key to working with linear equations. The slope describes how steep the line is and is represented by the letter \( m \). It is calculated as the change in the vertical direction (rise) over the change in the horizontal direction (run). A positive slope means the line ascends upward from left to right, while a negative slope descends. In our example, \( -\frac{1}{2} \) is the slope, which implies that for every 2 units you move to the right, the line drops down 1 unit. This visual movement helps clarify how the line progresses across the grid.
Y-Intercept
The y-intercept is a crucial component of the graph of a linear equation. It is the point where the line crosses the y-axis. Mathematically, it corresponds to the 'c' value in the slope-intercept form \( y = mx + c \). For the equation \( y = -\frac{1}{2}x + 2 \), the y-intercept is 2. This means the line will intersect the y-axis at the point (0, 2). Plotting this point accurately is the first step in graphing the equation. By identifying the y-intercept, you have an anchor point from which to plot additional points using the slope.
Plotting Points
Plotting points is an exciting step that brings your linear equation to life visually. After identifying the y-intercept, use the slope to determine another point on the line. Starting from the y-intercept, apply the rise-over-run concept to find a second point. For our slope of \(-\frac{1}{2}\), from point (0, 2), move down 1 unit and 2 units to the right, which lands you at the point (2, 1). By plotting this second point, you can accurately draw the line that represents the equation \(y = -\frac{1}{2}x + 2 \). The more accurately you plot these points, the clearer and more precise your line will be on the graph.

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Most popular questions from this chapter

In Exercises \(142-146,\) use the \([\mathrm{GRAPH}]\) or \([\text { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$4 x^{2}-12 x+9=(4 x-3)^{2} ;[-5,5,1] \text { by }[0,20,1]$$

Use factoring to solve quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\)-intercepts. \(25 x^{2}=49\)

Factor the numerator and the denominator. Then simplify by dividing out the common factor in the numerator and the denominator. \(\frac{x^{2}+6 x+5}{x^{2}-25}\)

In Exercises \(119-122\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\begin{array}{ccccccc}\text { All } & \text { perfect } & \text { square } & \text { trinomials } & \text { are } & \text { squares } & \text { of }\end{array}\) binomials.

Solve equation and check your solutions. \((x-3)^{2}+2(x-3)-8=0\)
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