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Mathematics LibreTexts

11.8: Understand Slope of a Line (Part 2)

  • Page ID
    5044
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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 11.40:

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

    Solution

    Plot the given point, (1, −1).

    The graph shows the x y-coordinate plane. The x-axis runs from -1 to 7. The y-axis runs from -3 to 4. A labeled point is drawn at “ordered pair 1, -1”.

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

    $$\begin{split} m &= \frac{3}{4} \\ \frac{rise}{run} &= \frac{3}{4} \\ rise &= 3 \\ run &= 4 \end{split}$$

    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.

    The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two line segments are drawn. A vertical line segment connects the points “ordered pair 1, -1” and “order pair “1, 2”. It is labeled “3”. A horizontal line segment starts at the top of the vertical line segment and goes to the right, connecting the points “ordered pair 1, 2” and “ordered pair 5, 2”. It is labeled “4”.

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

    The graph shows the x y-coordinate plane. The x-axis runs from -3 to 5. The y-axis runs from -1 to 7. Two unlabeled points are drawn at  “ordered pair 1, -1” and  “ordered pair 5, 2”.  A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair 1, -1” and “ordered pair 1, 2 ”.  A horizontal line segment connects “ordered pair 1, 2” and “ordered pair 5, 2”.

    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 11.78:

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

    (2, −2), m = \(\frac{4}{3}\)

    Exercise 11.79:

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

    (−2, 3), m = \(\frac{1}{4}\)

    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 11.41:

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

    Solution

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

    The graph shows the x y-coordinate plane. The x-axis runs from -1 to 4. The y-axis runs from -1 to 3. The point “ordered pair 0, 2” is labeled.

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

    $$\begin{split} m &= − \frac{2}{3} \\ \frac{rise}{run} &= \frac{−2}{3} \\ rise &= –2 \\ run &= 3 \end{split}$$

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

    The graph shows the x y-coordinate plane. Both axes run from -5 to 5. A vertical line segment connects points at “ordered pair 0, 2” and “ordered pair 0, 0” and is labeled “down 2”. A horizontal line segment connects “ordered pair 0, 0” and “ordered pair 0, 3” and is labeled “right 3”.

    Connect the points with a line.

    The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two labeled points are drawn at  “ordered pair 0, 2” and  “ordered pair 3, 0”.  A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair 0, 2” and “ordered pair 0, 0 ”.  A horizontal line segment connects “ordered pair 0, 0” and “ordered pair 3, 0”.

    Exercise 11.80:

    Graph the line with the given intercept and slope:

    y-intercept 4, m = \(− \frac{5}{2}\)

    Exercise 11.81:

    Graph the line with the given intercept and slope:

    x-intercept −3, m = \(− \frac{3}{4}\)

    Example 11.42:

    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. Both axes run from -5 to 5. The point “ordered pair -1, -3” is labeled.

    Identify the rise and the run. $$m = 4 \tag{11.4.44}$$
    Write 4 as a fraction. $$\frac{rise}{run} = \frac{4}{1} \tag{11.4.45}$$
    rise = 4, run = 1

    Count the rise and run.

    The graph shows the x y-coordinate plane. Both axes run from -5 to 5. The y-axis runs from -4 to 2. A vertical line segment connects points at “ordered pair -1,  -3” and “ordered pair -1, 1” and is labeled “up 4”. A horizontal line segment connects “ordered pair -1, 1” and “ordered pair 0, 1” and is labeled “over 1”.

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

    The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two labeled points are drawn at  “ordered pair -1, -3” and  “ordered pair -1, 1”.  A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair -1, -3” and “ordered pair -1, 1 ”. It is labeled “up 4” A horizontal line segment connects “ordered pair -1, 1” and “ordered pair 0, 1”. It is labeled “over 1”

    Exercise 11.82:

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

    Exercise 11.83:

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

    Solve Slope Applications

    At the beginning of this section, we said there are many applications of slope in the real world. Let’s look at a few now.

    Example 11.43:

    The pitch of a building’s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown?

    This figure shows a house with a sloped roof. The roof on one half of the building is labeled “pitch of the roof”. There is a line segment with arrows at each end measuring the vertical length of the roof and is labeled “rise = 9 feet”. There is a line segment with arrows at each end measuring the horizontal length of the root and is labeled “run = 18 feet”.

    Solution

    Use the slope formula. $$m = \frac{rise}{run} \tag{11.4.46}$$
    Substitute the values for rise and run. $$m = \frac{9\; ft}{18\; ft} \tag{11.4.47}$$
    Simplify. $$m = \frac{1}{2} \tag{11.4.48}$$
    The slope of the roof is \(\frac{1}{2}\).

    Exercise 11.84:

    Find the slope given rise and run: A roof with a rise = 14 and run = 24.

    Exercise 11.85:

    Find the slope given rise and run: A roof with a rise = 15 and run = 36.

    Have you ever thought about the sewage pipes going from your house to the street? Their slope is an important factor in how they take waste away from your house.

    Example 11.44:

    Sewage pipes must slope down \(\frac{1}{4}\) inch per foot in order to drain properly. What is the required slope?

    This figure shows a  right triangle. The short leg is vertical and is labeled “1 over 4 inch”. The long leg labeled “1 foot”.

    Solution

    Use the slope formula. $$m = \frac{rise}{run} = \frac{- \frac{1}{4}\; in.}{1\; ft} \tag{11.4.49}$$
    Convert 1 foot to 12 inches. $$m = \frac{- \frac{1}{4}\; in.}{12\; in} \tag{11.4.50}$$
    Simplify. $$m = - \frac{1}{48} \tag{11.4.51}$$
    The slope of the pipe is \(− \frac{1}{48}\).

    Exercise 11.86:

    Find the slope of the pipe: The pipe slopes down \(\frac{1}{3}\) inch per foot.

    Exercise 11.87:

    Find the slope of the pipe: The pipe slopes down \(\frac{3}{4}\) inch per yard.

    Practice Makes Perfect

    Use Geoboards to Model Slope

    In the following exercises, find the slope modeled on each geoboard.

    1. The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 5 row 2.
    2. The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 2 row 4 and the point in column 5 row 2.
    3. The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 2 row 1 and the point in column 4 row 4.
    4. The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 2 row 1 and the point in column 4 row 4.

    In the following exercises, model each slope. Draw a picture to show your results.

    1. \(\frac{2}{3}\)
    2. \(\frac{3}{4}\)
    3. \(\frac{1}{4}\)
    4. \(\frac{4}{3}\)
    5. \(- \frac{1}{2}\)
    6. \(- \frac{3}{4}\)
    7. \(- \frac{2}{3}\)
    8. \(- \frac{3}{2}\)

    Find the Slope of a Line from its Graph

    In the following exercises, find the slope of each line shown.

    1. The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, -4” and “ordered pair 10, 0”.
    2. The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, -5” and “ordered pair 3, 0”.
    3. The graph shows the x y-coordinate plane. The x-axis runs from -12 to 12. The y-axis runs from -12 to 12. A line passes through the points “ordered pair 0, -1” and “ordered pair 1, 0”.
    4. The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 3” and “ordered pair 6, 0”.
    5. The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 2” and “ordered pair 6, 0”.
    6. The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 2” and “ordered pair 6, 0”.
    7. The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 6” and “ordered pair 8, 0”.
    8. The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair -1,  0” and “ordered pair 0, -1”.
    9. The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair -4,  0” and “ordered pair -4, 6”.
    10. The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. The y-axis runs from -10 to 10. A line passes through the points “ordered pair -2,  0” and “ordered pair 4, 4”.
    11. The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 4” and “ordered pair 4, -6”.
    12. The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair 0, 4” and “ordered pair 4, -6”.
    13. The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair 1,  4” and “ordered pair 7, 0”.
    14. The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair 0,  3” and “ordered pair 7, 0”.
    15. The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair 2, 0” and “ordered pair 10, 4”.
    16. The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair 6,  2” and “ordered pair 0, -3”.

    Find the Slope of Horizontal and Vertical Lines

    In the following exercises, find the slope of each line.

    1. y = 3
    2. y = 1
    3. x = 4
    4. x = 2
    5. y = −2
    6. y = −3
    7. x = −5
    8. x = −4

    Use the Slope Formula to find the Slope of a Line between Two Points

    In the following exercises, use the slope formula to find the slope of the line between each pair of points.

    1. (1, 4), (3, 9)
    2. (2, 3), (5, 7)
    3. (0, 3), (4, 6)
    4. (0, 1), (5, 4)
    5. (2, 5), (4, 0)
    6. (3, 6), (8, 0)
    7. (−3, 3), (2, −5)
    8. (−2, 4), (3, −1)
    9. (−1, −2), (2, 5)
    10. (−2, −1), (6, 5)
    11. (4, −5), (1, −2)
    12. (3, −6), (2, −2)

    Graph a Line Given a Point and the Slope

    In the following exercises, graph the line given a point and the slope.

    1. (1, −2); m = \(\frac{3}{4}\)
    2. (1, −1); m = \(\frac{1}{2}\)
    3. (2, 5); m = \(− \frac{1}{3}\)
    4. (1, 4); m = \(− \frac{1}{2}\)
    5. (−3, 4); m = \(− \frac{3}{2}\)
    6. (−2, 5); m = \(− \frac{5}{4}\)
    7. . (−1, −4); m = \(\frac{4}{3}\)
    8. (−3, −5); m = \(\frac{3}{2}\)
    9. (0, 3); m = \(− \frac{2}{5}\)
    10. (0, 5); m = \(− \frac{4}{3}\)
    11. (−2, 0); m = \(−\frac{3}{4}\)
    12. (−1, 0); m = \(\frac{1}{5}\)
    13. (−3, 3); m = 2
    14. (−4, 2); m = 4
    15. (1, 5); m = −3
    16. (2, 3); m = −1

    Solve Slope Applications

    In the following exercises, solve these slope applications.

    1. Slope of a roof A fairly easy way to determine the slope is to take a 12-inch level and set it on one end on the roof surface. Then take a tape measure or ruler, and measure from the other end of the level down to the roof surface. You can use these measurements to calculate the slope of the roof. What is the slope of the roof in this picture?

    The figure shows a wood board at a diagonal representing a side-view slice of a pitched roof. A vertical line segment with arrows on both ends measures the vertical change in height of the roof and is labeled “4 inches”. A level tool is in a horizontal position above the board and above it is a line segment with arrows on both ends labeled “12 inches”.

    1. What is the slope of the roof shown?

    The figure shows a  diagonal side-view slice of a pitched roof. A ruler in vertical position is at the bottom of the roof segment and shows unit labels 1 through 8 and extends one further unit. A second ruler starts at the “7” label of the vertical ruler and extends horizontally until it hits the rising roof. The horizontal ruler has unit labels 1 through 11 and extends one further unit.

    1. Road grade A local road has a grade of 6%. The grade of a road is its slope expressed as a percent.
      1. Find the slope of the road as a fraction and then simplify the fraction.
      2. What rise and run would reflect this slope or grade?
    2. Highway grade A local road rises 2 feet for every 50 feet of highway.
      1. What is the slope of the highway?
      2. The grade of a highway is its slope expressed as a percent. What is the grade of this highway?

    Everyday Math

    1. Wheelchair ramp The rules for wheelchair ramps require a maximum 1 inch rise for a 12 inch run.
      1. How long must the ramp be to accommodate a 24-inch rise to the door?
      2. Draw a model of this ramp.
    2. Wheelchair ramp A 1-inch rise for a 16-inch run makes it easier for the wheelchair rider to ascend the ramp.
      1. How long must the ramp be to easily accommodate a 24-inch rise to the door?
      2. Draw a model of this ramp.

    Writing Exercises

    1. What does the sign of the slope tell you about a line?
    2. How does the graph of a line with slope m = \(\frac{1}{2}\) differ from the graph of a line with slope m = 2?
    3. Why is the slope of a vertical line undefined?
    4. Explain how you can graph a line given a point and its slope.

    Self Check

    (a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    CNX_BMath_Figure_AppB_069.jpg

    (b) On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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