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Chapter 3: Problem 85

Use slope-intercept graphing to graph the equation. $$ y=3 x-2 $$

Short Answer

Expert verified
Plot points at \( (0, -2) \) and \( (1, 1) \); draw a line through them.

Step by step solution

01

- Identify the Slope and Y-Intercept

The equation given is in slope-intercept form, which is \( y = mx + b \). In this equation, \( m \) represents the slope and \( b \) represents the y-intercept. For the equation \( y = 3x - 2 \), the slope \( m \) is 3 and the y-intercept \( b \) is -2.
02

- Plot the Y-Intercept

Start by plotting the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis. For \( b = -2 \), this point is \( (0, -2) \). Place a point on \( (0, -2) \) on the graph.
03

- Use the Slope to Find Another Point

The slope \( m \) indicates the rise over run. For \( m = 3 \), this means a rise of 3 units up for every 1 unit right. Starting at the y-intercept \( (0, -2) \), move 1 unit to the right to \( x = 1 \) and 3 units up to \( y = 1 \), plotting a second point at \( (1, 1) \).
04

- Draw the Line

With the points \( (0, -2) \) and \( (1, 1) \) plotted, draw a straight line through these points, extending it in both directions. This line represents the graph of the equation \( y = 3x - 2 \).

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Key Concepts

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

slope-intercept form
To understand slope-intercept form, start with the formula: \(y = mx + b\). This is called the slope-intercept form of a linear equation. Here, \(m\) denotes the slope - a measure of how steep the line is. The intercept \(b\) is the point where the line crosses the y-axis.
For example, in the equation \(y = 3x - 2\), \(m = 3\) and \(b = -2\). This tells us that the line has a slope of 3, meaning it rises three units for every run of one unit. The line crosses the y-axis at -2, which is its y-intercept.
Using the slope-intercept form makes graphing linear equations straightforward, as you can quickly identify both the slope and y-intercept.
plotting points
Plotting points is crucial for graphing linear equations. You start by determining key points where the line crosses the axes or other key locations. For the equation \(y = 3x - 2\), we begin by plotting the y-intercept. This is where the line intersects the y-axis. In this case, it's at point \(0, -2\).
Next, we use the slope to find additional points on the line. The slope \(m = 3\) indicates a rise of three units up for every one unit moved to the right. Starting at \(0, -2\), move one unit to the right (to \(x = 1\)) and three units up (to \(y = 1\)).
Plot this second point at \(1, 1\). Once these points are plotted, you can draw a line through them to visualize the equation.
slope
The slope of a line describes how steep it is. It's calculated as rise over run, or the change in y divided by the change in x (\(\frac{\text{rise}}{\text{run}} \)). In our equation \(y = 3x - 2\), the slope \(m\) is 3, meaning for every unit you move to the right on the x-axis, you move three units up on the y-axis.
Understanding slope helps you predict how the line behaves. A positive slope like 3 indicates an upward trend, while a negative slope would indicate a downward trend. If the slope were zero, the line would be horizontal, showing no change in y as x varies.
To use slope in graphing, start at a known point and use the rise/run ratio to find another point on the line.
y-intercept
The y-intercept is where the line crosses the y-axis. This happens when \(x = 0\). In slope-intercept form \(y = mx + b\), the y-intercept is represented by \(b\). For our equation \(y = 3x - 2\), the y-intercept is -2.
To plot the y-intercept, locate the point (0, -2) on the graph. This point is vital because it serves as the starting position for applying the slope. From the y-intercept, you use the slope to find other points on the line.
By marking the y-intercept first, you ensure that the line you draw is accurately positioned on the graph, from which you can extend using the slope.

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Most popular questions from this chapter

Use the slope formula to find the slope of the line that passes through the points. \((3,11) ;(7,39)\)

(a) write the equation of the horizontal line that passes through the point. (b) graph the equation. \((-2,-1)\)

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(y=-x+2\)

Use the slope formula to find the slope of the line that passes through the points. \((5,10) ;(7,16)\)

Use the slope formula to find the slope of the line that passes through the points. ( \(3 \mathrm{hr}, \$ 45) ;(7 \mathrm{hr}, \$ 105)\)
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