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Mathematics LibreTexts

4.4E: Exercises

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    30397
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    Practice Makes Perfect

    Use Geoboards to Model Slope

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

    Exercise \(\PageIndex{1}\)

    The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3 and the peg in column 5, row 2, forming a line.

    Answer

    \(\frac{1}{4}\)

    Exercise \(\PageIndex{2}\)

    The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 4 and the peg in column 5, row 2, forming a line.

    Exercise \(\PageIndex{3}\)

    The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 4, row 2, forming a line.

    Answer

    \(\frac{2}{3}\)

    Exercise \(\PageIndex{4}\)

    The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 3, row 4 and the peg in column 5, row 1, forming a line.

    Exercise \(\PageIndex{5}\)

    The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 1 and the peg in column 4, row 4, forming a line.

    Answer

    \(\frac{-3}{2}=-\frac{3}{2}\)

    Exercise \(\PageIndex{6}\)

    The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3 and the peg in column 5, row 4, forming a line.

    Exercise \(\PageIndex{7}\)

    The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 1 and the peg in column 5, row 4, forming a line.

    Answer

    \(-\frac{2}{3}\)

    Exercise \(\PageIndex{8}\)

    The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 2 and the peg in column 4, row 5, forming a line.

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

    Exercise \(\PageIndex{9}\)

    \(\frac{2}{3}\)

    Answer

    The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 5 and the peg in column 5, row 3, forming a line.

    Exercise \(\PageIndex{10}\)

    \(\frac{3}{4}\)

    Exercise \(\PageIndex{11}\)

    \(\frac{1}{4}\)

    Answer

    The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 5, row 3, forming a line.

    Exercise \(\PageIndex{12}\)

    \(\frac{4}{3}\)

    Exercise \(\PageIndex{13}\)

    \(-\frac{1}{2}\)

    Answer

    The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 3, row 5, forming a line.

    Exercise \(\PageIndex{14}\)

    \(-\frac{3}{4}\)

    Exercise \(\PageIndex{15}\)

    \(-\frac{2}{3}\)

    Answer

    The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2 and the peg in column 4, row 4, forming a line.

    Exercise \(\PageIndex{16}\)

    \(-\frac{3}{2}\)

    Use \(m=\frac{rise}{run}\) to find the Slope of a Line from its Graph

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

    Exercise \(\PageIndex{17}\)

    The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line passes through the points (negative 10, negative 8), (0, negative 4), and (10, 0).

    Answer

    \(\frac{2}{5}\)

    Exercise \(\PageIndex{18}\)

    The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, negative 8) and (2, negative 2).

    Exercise \(\PageIndex{19}\)

    The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 4, negative 6) and (4, 4).

    Answer

    \(\frac{5}{4}\)

    Exercise \(\PageIndex{20}\)

    The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line intercepts the y-axis at (0, negative 2) and passes through the point (3, 3).

    Exercise \(\PageIndex{21}\)

    The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 3, 3) and (3, 1).

    Answer

    \(-\frac{1}{3}\)

    Exercise \(\PageIndex{22}\)

    The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 4) and (2, 2).

    Exercise \(\PageIndex{23}\)

    The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line intercepts the y-axis at (0, 6) and passes through the point (4, 3).

    Answer

    \(-\frac{3}{4}\)

    Exercise \(\PageIndex{24}\)

    The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the point (negative 3, 1) and intercepts the y-axis at (0, negative 1).

    Exercise \(\PageIndex{25}\)

    The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 1) and (2, 4).

    Answer

    \(\frac{3}{4}\)

    Exercise \(\PageIndex{26}\)

    The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 1, 1) and (2, 3).

    Exercise \(\PageIndex{27}\)

    The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 1, 6) and (1, 1).

    Answer

    \(-\frac{5}{2}\)

    Exercise \(\PageIndex{28}\)

    The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the point (negative 1, 3) and intercepts the x-axis at (3, 0).

    Exercise \(\PageIndex{29}\)

    The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 6) and (1, 4).

    Answer

    \(-\frac{2}{3}\)

    Exercise \(\PageIndex{30}\)

    The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line passes through the points (negative 1, 3) and (1, 2).

    Exercise \(\PageIndex{31}\)

    The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 1).

    Answer

    \(\frac{1}{4}\)

    Exercise \(\PageIndex{32}\)

    The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line passes through the points (4, 2) and (7, 3).

    Find the Slope of Horizontal and Vertical Lines

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

    Exercise \(\PageIndex{33}\)

    y=3

    Answer

    0

    Exercise \(\PageIndex{34}\)

    y=1

    Exercise \(\PageIndex{35}\)

    x=4

    Answer

    undefined

    Exercise \(\PageIndex{36}\)

    x=2

    Exercise \(\PageIndex{37}\)

    y=−2

    Answer

    0

    Exercise \(\PageIndex{38}\)

    y=−3

    Exercise \(\PageIndex{39}\)

    x=−5

    Answer

    undefined

    Exercise \(\PageIndex{40}\)

    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.

    Exercise \(\PageIndex{41}\)

    (1,4),(3,9)

    Answer

    \(\frac{5}{2}\)

    Exercise \(\PageIndex{42}\)

    (2,3),(5,7)

    Exercise \(\PageIndex{43}\)

    (0,3),(4,6)

    Answer

    \(\frac{3}{4}\)

    Exercise \(\PageIndex{44}\)

    (0,1),(5,4)

    Exercise \(\PageIndex{45}\)

    (2,5),(4,0)

    Answer

    \(-\frac{5}{2}\)

    Exercise \(\PageIndex{46}\)

    (3,6),(8,0)

    Exercise \(\PageIndex{47}\)

    (−3,3),(4,−5)

    Answer

    \(-\frac{8}{7}\)

    Exercise \(\PageIndex{48}\)

    (−2,4),(3,−1)

    Exercise \(\PageIndex{49}\)

    (−1,−2),(2,5)

    Answer

    \(\frac{7}{3}\)

    Exercise \(\PageIndex{50}\)

    (−2,−1),(6,5)

    Exercise \(\PageIndex{51}\)

    (4,−5),(1,−2)

    Answer

    −1

    Exercise \(\PageIndex{52}\)

    (3,−6),(2,−2)

    Graph a Line Given a Point and the Slope

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

    Exercise \(\PageIndex{53}\)

    \((1,-2) ; m=\frac{3}{4}\)

    Answer

    The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (1, negative 2) and (5, 1).

    Exercise \(\PageIndex{54}\)

    \((1,-1) ; m=\frac{2}{3}\)

    Exercise \(\PageIndex{55}\)

    \((2,5) ; m=-\frac{1}{3}\)

    Answer

    The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (2, 5) and (5, 4).

    Exercise \(\PageIndex{56}\)

    \((1,4) ; m=-\frac{1}{2}\)

    Exercise \(\PageIndex{57}\)

    \((-3,4) ; m=-\frac{3}{2}\)

    Answer

    The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 3, 4) and (negative 1, 1).

    Exercise \(\PageIndex{58}\)

    \((-2,5) ; m=-\frac{5}{4}\)

    Exercise \(\PageIndex{59}\)

    \((-1,-4) ; m=\frac{4}{3}\)

    Answer

    The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 1, negative 4) and intercepts the x-axis at (2, 0).

    Exercise \(\PageIndex{60}\)

    \((-3,-5) ; m=\frac{3}{2}\)

    Exercise \(\PageIndex{61}\)

    \(y\) -intercept \(3 ; m=-\frac{2}{5}\)

    Answer

    The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the y-axis at (0, 3) and passes through the point (5, 1).

    Exercise \(\PageIndex{62}\)

    \(y\) -intercept \(5 ; m=-\frac{4}{3}\)

    Exercise \(\PageIndex{63}\)

    \(x\) -intercept \(-2 ; m=\frac{3}{4}\)

    Answer

    The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 3).

    Exercise \(\PageIndex{64}\)

    \(x\) -intercept \(-1 ; m=\frac{1}{5}\)

    Exercise \(\PageIndex{65}\)

    \((-3,3) ; m=2\)

    Answer

    The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (negative 3, 3) and (negative 2, 5).

    Exercise \(\PageIndex{66}\)

    \((-4,2) ; m=4\)

    Exercise \(\PageIndex{67}\)

    \((1,5) ; m=-3\)

    Answer

    The graph shows the x y coordinate plane. The x and y-axes run from negative 12 to 12. A line passes through the points (1, 5) and (2, 2).

    Exercise \(\PageIndex{68}\)

    \((2,3) ; m=-1\)

    Everyday Math

    Exercise \(\PageIndex{69}\)

    Slope of a roof. An easy way to determine the slope of a roof is to set one end of a 12 inch level on the roof surface and hold it level. Then take a tape measure or ruler and measure from the other end of the level down to the roof surface. This will give you the slope of the roof. Builders, sometimes, refer to this as pitch and state it as an “x 12 pitch” meaning \(\frac{x}{12}\), where x is the measurement from the roof to the level—the rise. It is also sometimes stated as an “x-in-12 pitch”.

    1. What is the slope of the roof in this picture?
    2. What is the pitch in construction terms?
      This figure shows one side of a sloped roof of a house. The rise of the roof is labeled “4 inches” and the run of the roof is labeled “12 inches”.
    Answer
    1. \(\frac{1}{3}\) 
    2. 4 12 pitch or 4-in-12 pitch

    Exercise \(\PageIndex{70}\)

    The slope of the roof shown here is measured with a 12” level and a ruler. What is the slope of this roof?

    This figure shows one side of a sloped roof of a house. The rise of the roof is measured with a ruler and shown to be 7 inches. The run of the roof is measured with a twelve inch level and shown to be 12 inches.

    Exercise \(\PageIndex{71}\)

    Road grade. A local road has a grade of 6%. The grade of a road is its slope expressed as a percent. Find the slope of the road as a fraction and then simplify. What rise and run would reflect this slope or grade?

    Answer

    \(\frac{3}{50} ;\) rise \(=3,\) run \(=50\)

    Exercise \(\PageIndex{72}\)

    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?

    Exercise \(\PageIndex{73}\)

    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. Create a model of this ramp.
    Answer
    1. 288 inches (24 feet) 
    2. Models will vary.

    Exercise \(\PageIndex{74}\)

    Wheelchair ramp. A 1-inch rise for a 16-inch run makes it easier for the wheelchair rider to ascend a ramp.

    1. How long must a ramp be to easily accommodate a 24-inch rise to the door?
    2. Create a model of this ramp.

    Writing Exercises

    Exercise \(\PageIndex{75}\)

    What does the sign of the slope tell you about a line?

    Answer

    When the slope is a positive number the line goes up from left to right. When the slope is a negative number the line goes down from left to right.

    Exercise \(\PageIndex{76}\)

    How does the graph of a line with slope \(m=\frac{1}{2}\) differ from the graph of a line with slope \(m=2 ?\)

    Exercise \(\PageIndex{77}\)

    Why is the slope of a vertical line “undefined”?

    Answer

    A vertical line has 0 run and since division by 0 is undefined the slope is undefined.

    Self Check

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

    This table has seven rows and four columns. The first row is a header row and it labels each column. The first column is labeled "I can …", the second "Confidently", the third “With some help” and the last "No–I don’t get it". In the “I can…” column the next row reads “use geoboards to model slope.” The third row reads “use m equals rise divided by run to find the slope of a line from its graph.” The fourth row reads “find the slope of horizontal and vertical lines.” The fifth row reads “use the slope formula to find the slope of a line between two points.” The sixth row reads “graph a line given a point and the slope.” The last row reads “solve slope applications.” The remaining columns are blank.

    ⓑ 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?

    Glossary

    geoboard
    A geoboard is a board with a grid of pegs on it.
    negative slope
    A negative slope of a line goes down as you read from left to right.
    positive slope
    A positive slope of a line goes up as you read from left to right.
    rise
    The rise of a line is its vertical change.
    run
    The run of a line is its horizontal change.
    slope formula
    The slope of the line between two points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) is \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\).
    slope of a line
    The slope of a line is \(m=\frac{\text { rise }}{\text { run }}\). The rise measures the vertical change and the run measures the horizontal change.