11.4 Understand slope of a line (Page 6/9)

Prealgebra

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

The graph shows the x y-coordinate plane. The x-axis runs from -7 to 7. The y-axis runs from -7 to 7. A line passes through the points “ordered pair -2, 1” and “ordered pair 0, 7”.

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

The graph shows the x y-coordinate plane. The x-axis runs from -7 to 7. The y-axis runs from -7 to 7. A line passes through the points “ordered pair 0, 6” and “ordered pair 0, 3”.

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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.

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.	$m = \frac{rise}{run}$
Substitute the values for rise and run.	$m = \frac{9 ft}{18 ft}$
Simplify.	$m = \frac{1}{2}$
	The slope of the roof is $\frac{1}{2}$ .

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

$\frac{7}{12}$

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

$\frac{5}{12}$

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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.

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.	$m = \frac{rise}{run}$
	$m = \frac{- \frac{1}{4} in .}{1 ft}$
	$m = \frac{- \frac{1}{4} in .}{1 ft}$
Convert 1 foot to 12 inches.	$m = \frac{- \frac{1}{4} in .}{12 in.}$
Simplify.	$m = - \frac{1}{48}$
	The slope of the pipe is $- \frac{1}{48} .$

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

$- \frac{1}{36}$

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

$- \frac{1}{48}$

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Access additional online resources

Key concepts

Find the slope from a graph
1. Locate two points on the line whose coordinates are integers.
2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
3. Count the rise and the run on the legs of the triangle.
4. Take the ratio of rise to run to find the slope, $m = \frac{rise}{run}$
Slope of a Horizontal Line
- The slope of a horizontal line, $y = b$ , is 0.
Slope of a Vertical Line
- The slope of a vertical line, $x = a$ , is undefined.
Slope Formula
- The slope of the line between two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$
Graph a line given a point and a slope.
1. Plot the given point.
2. Use the slope formula to identify the rise and the run.
3. Starting at the given point, count out the rise and run to mark the second point.
4. Connect the points with a line.

Practice makes perfect

Use Geoboards to Model Slope

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

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.

$\frac{1}{4}$

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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.

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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.

$- \frac{3}{2}$

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

$\frac{2}{3}$

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 5 and the point in column 5 row 3.

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$\frac{3}{4}$

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$\frac{1}{4}$

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 1 row 4 and the point in column 5 row 3.

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$\frac{4}{3}$

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$- \frac{1}{2}$

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 1 row 4 and the point in column 3 row 5.

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$- \frac{3}{4}$

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$- \frac{2}{3}$

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 1 row 2 and the point in column 4 row 4.

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$- \frac{3}{2}$

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Find the Slope of a Line from its Graph

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

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”.

$\frac{2}{5}$

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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”.

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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”.

$\frac{5}{4}$

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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”.

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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”.

$- \frac{1}{3}$

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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”.

$- \frac{3}{4}$

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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”.

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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”.

$\frac{3}{4}$

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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”.

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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”.

$- \frac{5}{2}$

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The graph shows the x y-coordinate plane. The x-axis runs from -10 to 10. A line passes through the points “ordered pair -8, 8” and “ordered pair 8, -4”.

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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”.

$- \frac{2}{3}$

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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”.

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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”.

$\frac{1}{4}$

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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”.

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Find the Slope of Horizontal and Vertical Lines

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

$y = 3$

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$y = 1$

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$x = 4$

undefined

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$x = 2$

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$y = −2$

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$y = −3$

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$x = −5$

undefined

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$x = −4$

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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.

$(1, 4), (3, 9)$

$\frac{5}{2}$

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$(2, 3), (5, 7)$

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$(0, 3), (4, 6)$

$\frac{3}{4}$

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$(0, 1), (5, 4)$

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$(2, 5), (4, 0)$

$- \frac{5}{2}$

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$(3, 6), (8, 0)$

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Source: OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9

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