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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:

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