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The slope of a line

Algebra, grade 8 Lesson 10 of 27Learning

The slope of a line

The slope of a line is the rate of change of its function: how much y\displaystyle y changes when x\displaystyle x grows by 1. It is written with the letter m\displaystyle m.

  • The rate of a linear function is constant, so a line has one slope – everywhere on it.
  • The slope shows which way the line goes and how steep it is.
The slope from two points
You may start from either point, as long as you start from it in both the numerator and the denominator: 3−91−4=−6−3=2\displaystyle {\frac{3-9}{1-4}}={\frac{-6}{-3}}={2}.

Between two points on a line, divide the change in y\displaystyle y by the change in x\displaystyle x:

m=y2−y1x2−x1\displaystyle m=\frac{y_2-y_1}{x_2-x_1}

(x1,y1)\displaystyle (x_1,y_1) is one point and (x2,y2)\displaystyle (x_2,y_2) is the other.

y=2x+1\displaystyle y=2x+1
A=(1,3)\displaystyle A=(1,3)
B=(4,9)\displaystyle B=(4,9)

The slope of the line through A(1,3)\displaystyle A(1,3) and B(4,9)\displaystyle B(4,9):

  • The change in y\displaystyle y: 9−3=6\displaystyle {9-3}={6}.
  • The change in x\displaystyle x: 4−1=3\displaystyle {4-1}={3}.
  • m=63=2\displaystyle {m}={\frac{6}{3}}={2}: at every step of 1 to the right, the line goes up 2.
A positive, a negative and a zero slope
A vertical line, like x=2\displaystyle {x}={2}, has no slope: the change in x\displaystyle x is 0, and you cannot divide by 0. A vertical line is not the graph of a function either.

The sign of the slope tells which way the line goes, read from left to right:

  • A positive slope – the line goes up. y=2x+1\displaystyle {y}={2x+1} has the slope 2\displaystyle 2.
  • A negative slope – the line goes down. y=−12x+2\displaystyle {y}={-\frac{1}{2}x+2} has the slope −12\displaystyle -\frac{1}{2}: at every step to the right it goes down a half.
  • A slope of 0 – the line is horizontal. In y=3\displaystyle {y}={3} all the points are at the same height, and y\displaystyle y does not change.
y=2x+1\displaystyle y=2x+1
y=−12x+2\displaystyle y=-\frac{1}{2}x+2
y=3\displaystyle y=3
Slope and steepness

The further the slope is from 0, the steeper the line.

  • y=13x\displaystyle {y}={\frac{1}{3}x}: slope 13\displaystyle \frac{1}{3} – it rises slowly.
  • y=x\displaystyle {y}={x}: slope 1.
  • y=3x\displaystyle {y}={3x}: slope 3 – steep.
  • y=−3x\displaystyle {y}={-3x} (dashed): slope −3\displaystyle -3 – just as steep, but going down.
y=13x\displaystyle y=\frac{1}{3}x
y=x\displaystyle y=x
y=3x\displaystyle y=3x
y=−3x\displaystyle y=-3x
Parallel lines

Lines with the same slope are parallel. They rise at the same rate, so the vertical distance between them does not change and they never meet.

  • y=2x\displaystyle {y}={2x}, y=2x+3\displaystyle {y}={2x+3} and y=2x−2\displaystyle {y}={2x-2} have the same slope, 2, and they are parallel.
  • The other way round too: parallel lines (that are not vertical) have the same slope.
y=2x\displaystyle y=2x
y=2x+3\displaystyle y=2x+3
y=2x−2\displaystyle y=2x-2
Exercise 1

Work out the slope of the line through the two points.
For each line, write whether it goes up, goes down or is horizontal.

a
(2,1)\displaystyle (2,1), (5,7)\displaystyle (5,7)

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b
    (−1,4)\displaystyle (-1,4), (3,−4)\displaystyle (3,-4)

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      c
      (0,5)\displaystyle (0,5), (6,8)\displaystyle (6,8)

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        d
        (−3,2)\displaystyle (-3,2), (4,2)\displaystyle (4,2)

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          Exercise 2

          Find the slope of each line from the graph.
          On each line pick two points where it passes exactly through the corner of a square.

          f(x)= ?\displaystyle f(x)=\,?
          g(x)= ?\displaystyle g(x)=\,?
          f\displaystyle f

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            g\displaystyle g

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              Exercise 3

              Work out the slopes of the lines AB\displaystyle AB, CD\displaystyle CD and EF\displaystyle EF.
              Which of them are parallel?

              A=(0,1)\displaystyle A=(0,1)
              B=(2,5)\displaystyle B=(2,5)
              C=(1,−2)\displaystyle C=(1,-2)
              D=(3,2)\displaystyle D=(3,2)
              E=(−1,4)\displaystyle E=(-1,4)
              F=(2,−2)\displaystyle F=(2,-2)
              AB\displaystyle AB

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                CD\displaystyle CD

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                  EF\displaystyle EF

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                    Exercise 4

                    A line passes through the point (1,2)\displaystyle (1,2).
                    Its slope is 3.

                    • Find two more points on the line.
                    • Is the point (4,11)\displaystyle (4,11) on the line? Explain.