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Intro to slope

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 left parenthesis, 3, comma, 2, right parenthesis and left parenthesis, 5, comma, 8, right parenthesis as an example:
The slope of a line describes how steep a line is. Slope is the change in y values divided by the change in x values.
Let's find the slope of the line that goes through the points left parenthesis, 3, comma, 2, right parenthesis and left parenthesis, 5, comma, 8, right parenthesis:
start text, S, l, o, p, e, end text, equals, start fraction, start color #e07d10, start text, C, h, a, n, g, e, space, i, n, space, y, end text, end color #e07d10, divided by, start color #1fab54, start text, C, h, a, n, g, e, space, i, n, space, x, end text, end color #1fab54, end fraction, equals, start fraction, start color #e07d10, 6, end color #e07d10, divided by, start color #1fab54, 2, end color #1fab54, end fraction, equals, 3
Use the graph below to find the slope of the line that goes through the points left parenthesis, 1, comma, 2, right parenthesis and left parenthesis, 6, comma, 6, right parenthesis.
start text, S, l, o, p, e, end text, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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 left parenthesis, 2, comma, 7, right parenthesis and left parenthesis, 5, comma, 1, right parenthesis.
start text, S, l, o, p, e, end text, equals, start fraction, start color #e07d10, start text, C, h, a, n, g, e, space, i, n, space, y, end text, end color #e07d10, divided by, start color #1fab54, start text, C, h, a, n, g, e, space, i, n, space, x, end text, end color #1fab54, end fraction, equals, start fraction, start color #e07d10, minus, 6, end color #e07d10, divided by, start color #1fab54, 3, end color #1fab54, end fraction, equals, minus, 2
Wait a minute! Did you catch that? The change in y values is negative because we went from 7 down to 1. 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 left parenthesis, 1, comma, 9, right parenthesis and left parenthesis, 4, comma, 0, right parenthesis.
start text, S, l, o, p, e, end text, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Slope as "rise over run"

A lot of people remember slope as "rise over run" because slope is the "rise" (change in y) divided by the "run" (change in x).
start text, S, l, o, p, e, end text, equals, start fraction, start color #e07d10, start text, C, h, a, n, g, e, space, i, n, space, y, end text, end color #e07d10, divided by, start color #1fab54, start text, C, h, a, n, g, e, space, i, n, space, x, end text, end color #1fab54, end fraction, equals, start fraction, start color #e07d10, start text, R, i, s, e, end text, end color #e07d10, divided by, start color #1fab54, start text, R, u, n, end text, end color #1fab54, end fraction

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 left parenthesis, 7, comma, 4, right parenthesis and left parenthesis, 3, comma, 2, right parenthesis.
start text, S, l, o, p, e, end text, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

2) Use the graph below to find the slope of the line that goes through the points left parenthesis, minus, 6, comma, 9, right parenthesis and left parenthesis, 2, comma, 1, right parenthesis.
start text, S, l, o, p, e, end text, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

3) Use the graph below to find the slope of the line that goes through the points left parenthesis, minus, 8, comma, minus, 3, right parenthesis and left parenthesis, 4, comma, minus, 6, right parenthesis.
start text, S, l, o, p, e, end text, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

4) Use the graph below to find the slope of the line that goes through the points left parenthesis, 4, comma, 5, right parenthesis and left parenthesis, 9, comma, 5, right parenthesis.
start text, S, l, o, p, e, end text, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

5) Use the graph below to choose the slope of the line that goes through the points left parenthesis, 3, comma, 2, right parenthesis and left parenthesis, 3, comma, 8, right parenthesis.
start text, S, l, o, p, e, end text, equals
Choose 1 answer:
Choose 1 answer:

Challenge problems

See how well you understand slope by trying a couple of true/false problems.
6) A line with a slope of 5 is steeper than a line with a slope of start fraction, 1, divided by, 2, end fraction
Choose 1 answer:
Choose 1 answer:

7) A line with a slope of minus, 5 is steeper than a line with a slope of minus, start fraction, 1, divided by, 2, end fraction
Choose 1 answer:
Choose 1 answer:

Want to join the conversation?

  • orange juice squid orange style avatar for user Charlie Chan
    How can the slope value (1/2 or 5) be used in real life, and how can we use it in math?

    Thanks!
    (32 votes)
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  • blobby green style avatar for user Tama Haydock
    why does math have to be so confusing?
    (21 votes)
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  • old spice man green style avatar for user Mia
    My dad left to get milk. He never came back :)
    (18 votes)
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  • blobby green style avatar for user hyenzmacababbad
    Can somebody tell me how to easily visualized. which slope is steeper?
    (7 votes)
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    • starky tree style avatar for user Ina
      Try to think about it like this, imagine you are running up (positive slope) or down (negative slope) a flight of stairs.

      If the slope is a larger number, than it is the same as taking several steps at once.
      If the slope is a smaller number, it is as if you are taking less steps at once, not going up the flight of stairs as quickly.

      Imagine a slope of 1, means for ever step you take with your feet you only go up one stair.

      Imagine a slope of 5, this means for every step you take you go up 5 stairs. You get to the top and rise much quicker.

      ___________

      Another visual example: Imagine you are going skiing. As you go down a slope, you expect the slope to be negative. You come from up high on the y axis and go down.

      If now you go to a ski slope at -5 that means for every meter you glide forward on your skis towards to bottom of the hill (x-axis), you also go down 5 meters on the hills height (y-axis). This is very steep as you can imagine.

      Now imagine you are going down a ski slope of only 1/2. This means for every 1 meter you glide forward on your ski you only get 1/2 meter further down the hills height.
      (15 votes)
  • duskpin ultimate style avatar for user Ingrid
    Wait, if it's based on absolute value, how come in the video Slope & direction of a line, it was that the lesser negative was greater?
    (10 votes)
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    • hopper cool style avatar for user Seed Something
      In the Slope and Direction video we were determining a slope's direction on the number line, its location, and using values of specific numbers, so we were using the number line, (and smaller negative numbers have a greater value, as they are closer to zero and positive values).

      But…
      ★for Steepness we're no longer comparing direction or location, we're now comparing how vertical a line is to another.

      By comparing the absolute values of the slopes we can find which line has a greater change in y than the relative change to x.

      Find which line increases or decreases further and faster on the y-axis than another line.

      So by finding which is furthest from zero, the one with the greater absolute value, we know which line is steeper and had a more extreme change in y than the other line.

      (ㆁωㆁ) Hope this helps someone!
      (0 votes)
  • blobby green style avatar for user jonny.savois
    lmao what is y=mx+b gonna do for me in life
    (8 votes)
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  • duskpin tree style avatar for user Naomi-Ife Lagbara
    Is there another way of finding a slope without a graph?
    (4 votes)
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  • starky tree style avatar for user chris ruiz
    would you use this in the field of archaeology
    (7 votes)
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  • duskpin seed style avatar for user yanet.sanchez1207
    is there an easier way to find the slope?
    (3 votes)
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    • starky tree style avatar for user Haleen
      yes, actually if you align the coordinates
      For example: (3,2)
      (5,8)
      And then you would find the difference between the the numbers such as the difference between the 3 and 5 is 2, while the difference in the 2 and 8 is 6. After you get these numbers you can put them in the fraction form as you would after you find the numbers on the graph.This would then equal 6/2 or in simplified form = 3.
      I hope this was helpful (:
      (9 votes)
  • aqualine ultimate style avatar for user Chinmay Joshi
    What does steeper mean, greater slope? In that case, shouldn't a slope of -1/2 be steeper than -5, as -1/2 is greater than -5?
    (3 votes)
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    • aqualine ultimate style avatar for user Kayla Aurellia
      We are seeing absolute value, which means if it's negative, we have to see the more negative one as the greater one. If it's positive, we have to see the more positive one as the greater one(like we always did)
      The symbol for absolute values are like so: | insert number |

      Try these problems as examples:
      Put > < = in the blanks.
      a) | -6 | _ | -1 |
      b) | 8 | _
      | 9 |
      c) | -4 | __ | -12 |

      The answers are
      a) > b) < c) <
      (7 votes)