Slope

Intro to slope

CCSS Math: 8.F.B.4
Walk through a graphical explanation of how to find the slope from two points and what it means.
We can draw a line through any two points on the coordinate plane.
Let's take the points (3,2)(3,2) and (5,8)(5, 8) as an example:
The slope of a line describes how steep a line is. Slope is the change in yy values divided by the change in xx values.
Let's find the slope of the line that goes through the points (3,2)(3,2) and (5,8)(5, 8):
Slope=Change in yChange in x=62=3\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD6}{\greenD 2} = 3
Use the graph below to find the slope of the line that goes through the points (1,2)(1,2) and (6,6)(6,6).
Slope=\text{Slope} =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

[Show solution]
Slope=Change in yChange in x=45\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD4}{\greenD 5}
Notice that both of the lines we've looked at so far have been increasing and have had positive slopes as a result. Now let's find the slope of a decreasing line.

Negative slope

Let's find the slope of the line that goes through the points (2,7)(2,7) and (5,1)(5, 1).
Slope=Change in yChange in x=−63=−2\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD{-6}}{\greenD3} = -2
Wait a minute! Did you catch that? The change in yy values is negative because we went from 77 down to 11. This led to a negative slope, which makes sense because the line is decreasing.
Use the graph below to find the slope of the line that goes through the points (1,9)(1,9) and (4,0)(4,0).
Slope=\text{Slope} =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

[Show solution]
Slope=Change in yChange in x=−93=−3\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD{-9}}{\greenD 3} = -3

Slope as "rise over run"

A lot of people remember slope as "rise over run" because slope is the "rise" (change in yy) divided by the "run" (change in xx).
Slope=Change in yChange in x=RiseRun\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}=\dfrac{\goldD{\text{Rise}}}{{\greenD{\text{Run}}}}

Let's practice!

Heads up! All of the examples we've seen so far have been points in the first quadrant, but that won't always be the case in the practice problems.
1) Use the graph below to find the slope of the line that goes through the points (7,4)(7,4) and (3,2)(3,2).
Slope=\text{Slope} =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

[Show solution]
Slope=Change in yChange in x=24=12\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD2}{\greenD 4} = \dfrac12
2) Use the graph below to find the slope of the line that goes through the points (−6,9)(-6,9) and (2,1)(2,1).
Slope=\text{Slope} =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

[Show solution]
Slope=Change in yChange in x=−88=−1\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD{-8}}{\greenD 8} = -1
3) Use the graph below to find the slope of the line that goes through the points (−8,−3)(-8,-3) and (4,−6)(4,-6).
Slope=\text{Slope} =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

[Show solution]
Slope=Change in yChange in x=−312=−14\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD{-3}}{\greenD {12}} = \dfrac{-1}4
4) Use the graph below to find the slope of the line that goes through the points (4,5)(4,5) and (9,5)(9,5).
Slope=\text{Slope} =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

[Show solution]
This is a tricky one because there is 0\goldD 0 change in yy.
Slope=Change in yChange in x=05=0\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD0}{\greenD 5} = 0
In fact, the slope of a horizontal line is always 00.
5) Use the graph below to choose the slope of the line that goes through the points (3,2)(3,2) and (3,8)(3,8).
Slope=\text{Slope} =
Choose 1 answer:
Choose 1 answer:

[Show solution]
This is a tricky one because there is 0\greenD 0 change in xx.
Slope=Change in yChange in x=60=\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD6}{\greenD 0} = undefined
Remember that we can't divide by 00, which is why the slope is undefined.
In fact, the slope of a vertical line is always undefined. Interestingly, sometimes the slope of a vertical line is also said to be infinity.

Challenge problems

See how well you understand slope by trying a couple of true/false problems.
6) A line with slope of 55 is steeper than a line with a slope of 12\dfrac12
Choose 1 answer:
Choose 1 answer:

[Show solution]
True! For increasing lines, lines with greater slopes have greater changes in yy relative to the same change in xx.
7) A line with slope of −5-5 is steeper than a line with a slope of −12-\dfrac12
Choose 1 answer:
Choose 1 answer:

[Show solution]
True! This is a bit tricky though! The steepness of a line is determined by the absolute value of its slope.
A line with a slope of −5\blueD{-5} is steeper than a line with a slope of −12\maroonD{-\dfrac12} because ∣−5∣|\blueD{-5}| is greater than ∣−12∣|\maroonD{-\dfrac12}|.
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