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Graph a Line Given a Point and the Slope

In this chapter, we graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.

Another method we can use to graph lines is the point-slope method. Sometimes, we will be given one point and the slope of the line, instead of its equation. When this happens, we use the definition of slope to draw the graph of the line.

Example (PageIndex{11}):

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

Solution

Plot the given point, (1, −1).

Use the slope formula m = (dfrac{rise}{run}) to identify the rise and the run.

[begin{split} m &= dfrac{3}{4} dfrac{rise}{run} &= dfrac{3}{4} rise &= 3 run &= 4 end{split}]

Starting at the point we plotted, count out the rise and run to mark the second point. We count 3 units up and 4 units right.

Then we connect the points with a line and draw arrows at the ends to show it continues.

We can check our line by starting at any point and counting up 3 and to the right 4. We should get to another point on the line.

Exercise (PageIndex{21}):

Graph the line passing through the point with the given slope:

(2, −2), m = (dfrac{4}{3})

Answer

Exercise (PageIndex{22}):

Graph the line passing through the point with the given slope:

(−2, 3), m = (dfrac{1}{4})

Answer

HOW TO: GRAPH A LINE GIVEN A POINT AND A SLOPE

Step 1. Plot the given point.

Step 2. Use the slope formula 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 (PageIndex{12}):

Graph the line with y -intercept (0, 2) and slope m = (− dfrac{2}{3}).

Solution

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

Use the slope formula m = rise run to identify the rise and the run.

[begin{split} m &= − dfrac{2}{3} dfrac{rise}{run} &= dfrac{−2}{3} rise &= –2 run &= 3 end{split}]

Starting at (0, 2), count the rise and the run and mark the second point.

Connect the points with a line.

Exercise (PageIndex{23}):

Graph the line with the given intercept and slope:

y-intercept 4, m = (− dfrac{5}{2})

Answer

Exercise (PageIndex{24}):

Graph the line with the given intercept and slope:

x-intercept −3, m = (− dfrac{3}{4})

Answer

Example (PageIndex{13}):

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

Solution

Plot the given point.

Count the rise and run.

Mark the second point. Connect the two points with a line.

Exercise (PageIndex{25}):

Graph the line with the given intercept and slope: (−2, 1), m = 3.

Answer

Exercise (PageIndex{26}):

Graph the line with the given intercept and slope: (4, −2), m = −2.

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Answer

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