Understanding the Slope of a Line

Graph a Line Given a Point and the Slope

Up to now, in this chapter, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.

One other method we can use to graph lines is called the point–slope method. We will use this method when we know one point and the slope of the line. We will start by plotting the point and then use the definition of slope to draw the graph of the line.

Example 4.35

How To Graph a Line Given a Point and The Slope

Graph the line passing through the point \((1,−1)\) whose slope is \(m=\frac{3}{4}\).

Solution

Step 1. Plot the given point.	Plot \((1,-1)\)
Step 2. Use the slope formula \(m=\frac{\text { rise }}{\text { run }}\) to identify the rise and the run.	Identify the rise and the run.	\(\begin{aligned} m &=\frac{3}{4} \\ \frac{\text { rise }}{\text { run }} &=\frac{3}{4} \\ \text { rise } &=3 \\ \text { run } &=4 \end{aligned}\)
Step 3. Starting at the given point, count out the rise and run to mark the second point.	Start at \((1,-1)\) and count the rise and the run. Up 3 units, right 4 units.
Step 4. Connect the points with a line.	Connect the two points with a line.

Try It 4.69

Graph the line passing through the point \((2, −2)\) with the slope \(m=\frac{4}{3}\).

Try It 4.70

Graph the line passing through the point \((−2,3)\) with the slope \(m=\frac{1}{4}\).

HOW TO

Graph a line given a point and the slope.

Step 1. Plot the given point.
Step 2. Use the slope formula \(m=\frac{\text { rise }}{\text { run }}\) to identify the rise and the run.
Step 3. Starting at the given point, count out the rise and run to mark the second point.
Step 4. Connect the points with a line.

Example 4.36

Graph the line with \(y\)-intercept 2 whose slope is \(m=−\frac{2}{3}\).

Solution

Plot the given point, the y-intercept, \((0, 2)\).

The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The point (0, 2) is plotted.

Identify the rise and the run.	\(m= −\frac{2}{3}\)
	\(\frac{rise}{ run} =\frac{−2}{3}\)
	\(rise =−2\)
	\(run =3\)

Table 4.38

Count the rise and the run. Mark the second point.

The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The points (0, 2), (0, 0), and (3,0) are

Connect the two points with a line.

The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. A line passes through the plotted points

You can check your work by finding a third point. Since the slope is \(m = - \frac{2}{3}\), it can be written as \(m = \frac{2}{-3}\). Go back to \((0, 2)\) and count out the rise, 2, and the run, -3.

Try It 4.71

Graph the line with the \(y\)-intercept \(4\) and slope \(m=-\frac{5}{2}\).

Try It 4.72

Graph the line with the \(x\)-intercept \(−3\) and slope \(m=-\frac{3}{4}\).

Example 4.37

Graph the line passing through the point \((−1,−3)\) whose slope is \(m=4\).

Solution

Plot the given point.

The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The point (negative 1, negative 3) is pl

Identify the rise and the run.	\(m=4\)
Write 4 as a fraction.	\(\frac{\text { rise }}{\text { run }}=\frac{4}{1}\)
	\(rise=4\), \(run=1\)

Table 4.39

Count the rise and run and mark the second point.

This figure shows how to graph the line passing through the point (negative 1, negative 3) whose slope is 4. The first step i

Connect the two points with a line.

The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. A line passes through the plotted points

You can check your work by finding a third point. Since the slope is \(m=4\), it can be written as \(m=\frac{-4}{-1} \). Go back to \((-1,-3)\) and count out the rise, \(-4\), and the run, \(-1\).

Try It 4.73

Graph the line with the point \((−2,1)\) and slope \(m=3\).

Try It 4.74

Graph the line with the point \((4,−2)\) and slope \(m=−2\).