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Linear functions: straight-line graphs

Algebra, grade 8 Lesson 8 of 27Learning

Linear functions: straight-line graphs

A linear function is a function whose graph is a straight line.

  • y=2x\displaystyle {y}={2x}, y=x+3\displaystyle {y}={x+3} and y=−x+4\displaystyle {y}={-x+4} are linear functions.
  • y=x2\displaystyle {y}={x^{2}} is not a linear function: its graph curves.
The function y=2x\displaystyle {y}={2x}
At every step to the right in the table x\displaystyle x grows by 1, and y\displaystyle y by 2.

In y=2x\displaystyle {y}={2x} every value of y\displaystyle y is 2 times its x\displaystyle x:

x\displaystyle x−2\displaystyle -2−1\displaystyle -10\displaystyle 01\displaystyle 12\displaystyle 23\displaystyle 3
y\displaystyle y−4\displaystyle -4−2\displaystyle -20\displaystyle 02\displaystyle 24\displaystyle 46\displaystyle 6

Each column of the table is a point on the coordinate plane, and all the points lie on one straight line.

y=2x\displaystyle y=2x
(−2,−4)\displaystyle (-2,-4)
(−1,−2)\displaystyle (-1,-2)
(0,0)\displaystyle (0,0)
(1,2)\displaystyle (1,2)
(2,4)\displaystyle (2,4)
(3,6)\displaystyle (3,6)

Every point between the table’s points is on the line too: x=12\displaystyle {x}={\frac{1}{2}} gives y=1\displaystyle {y}={1}, and the point (12,1)\displaystyle (\frac{1}{2},1) is on the line. So the points are joined by a line.

The function y=x+3\displaystyle {y}={x+3}

In y=x+3\displaystyle {y}={x+3} every value of y\displaystyle y is 3 more than its x\displaystyle x:

x\displaystyle x−3\displaystyle -3−2\displaystyle -2−1\displaystyle -10\displaystyle 01\displaystyle 12\displaystyle 2
y\displaystyle y0\displaystyle 01\displaystyle 12\displaystyle 23\displaystyle 34\displaystyle 45\displaystyle 5

At every step to the right y\displaystyle y grows by 1.

y=x+3\displaystyle y=x+3
y=x\displaystyle y=x

The dashed line is the graph of y=x\displaystyle {y}={x}. Every point of y=x+3\displaystyle {y}={x+3} is 3 higher than the point of y=x\displaystyle {y}={x} below it, so the two lines are parallel.

The function y=−x+4\displaystyle {y}={-x+4}

In y=−x+4\displaystyle {y}={-x+4} you work out 4\displaystyle 4 minus x\displaystyle x:

x\displaystyle x−1\displaystyle -10\displaystyle 01\displaystyle 12\displaystyle 23\displaystyle 34\displaystyle 4
y\displaystyle y5\displaystyle 54\displaystyle 43\displaystyle 32\displaystyle 21\displaystyle 10\displaystyle 0

At every step to the right y\displaystyle y gets 1 smaller, so the line goes down from left to right.

y=−x+4\displaystyle y=-x+4
Equal steps in the table
The steps of x\displaystyle x must be equal: if x\displaystyle x jumps by 1 once and by 2 the next time, the steps of y\displaystyle y differ too.

In the table of a linear function, when x\displaystyle x grows in equal steps, y\displaystyle y changes by the same number at every step.

  • In y=2x\displaystyle {y}={2x} – it grows by 2 at every step.
  • In y=x+3\displaystyle {y}={x+3} – it grows by 1.
  • In y=−x+4\displaystyle {y}={-x+4} – it gets 1 smaller.

In y=x2\displaystyle {y}={x^{2}} it is not so:

x\displaystyle x0\displaystyle 01\displaystyle 12\displaystyle 23\displaystyle 34\displaystyle 4
y\displaystyle y0\displaystyle 01\displaystyle 14\displaystyle 49\displaystyle 916\displaystyle 16

y\displaystyle y grows by 1, then by 3, by 5 and by 7 – more at every step. So its graph curves upwards.

y=x2\displaystyle y=x^{2}
Exercise 1

Which of the tables can be the table of a linear function?
For those tables, write by how much y\displaystyle y changes at every step.

a.

x\displaystyle x0\displaystyle 01\displaystyle 12\displaystyle 23\displaystyle 34\displaystyle 4
y\displaystyle y1\displaystyle 14\displaystyle 47\displaystyle 710\displaystyle 1013\displaystyle 13

b.

x\displaystyle x0\displaystyle 01\displaystyle 12\displaystyle 23\displaystyle 34\displaystyle 4
y\displaystyle y0\displaystyle 01\displaystyle 13\displaystyle 36\displaystyle 610\displaystyle 10

c.

x\displaystyle x−2\displaystyle -2−1\displaystyle -10\displaystyle 01\displaystyle 12\displaystyle 2
y\displaystyle y8\displaystyle 86\displaystyle 64\displaystyle 42\displaystyle 20\displaystyle 0

d.

x\displaystyle x0\displaystyle 02\displaystyle 24\displaystyle 46\displaystyle 68\displaystyle 8
y\displaystyle y5\displaystyle 56\displaystyle 67\displaystyle 78\displaystyle 89\displaystyle 9
A point on the line

A point is on the graph of a function when it satisfies the function’s formula. We substitute the point’s x\displaystyle x-coordinate in the formula, and check whether its y\displaystyle y-coordinate comes out.
We check two points against the line y=2x+1\displaystyle {y}={2x+1}:

  • A(3,7)\displaystyle A(3,7): 2⋅3+1=7\displaystyle {2\cdot3+1}={7}. It comes out 7, so A\displaystyle A is on the line.
  • B(4,10)\displaystyle B(4,10): 2⋅4+1=9\displaystyle {2\cdot4+1}={9}. It comes out 9, not 10, so B\displaystyle B is not on the line: it is above it.
y=2x+1\displaystyle y=2x+1
A=(3,7)\displaystyle A=(3,7)
B=(4,10)\displaystyle B=(4,10)
Exercise 2

Which of the points are on the line y=3x−2\displaystyle {y}={3x-2}?
For each point, work out the formula’s value and compare it with the point’s y\displaystyle y-coordinate.

a
(2,4)\displaystyle (2,4)

    / – 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

    b
    (−1,−5)\displaystyle (-1,-5)

      / – 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
      (0,2)\displaystyle (0,2)

        / – 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

        d
        (5,12)\displaystyle (5,12)

          / – 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

          Complete a table of the function y=−2x+3\displaystyle {y}={-2x+3} for x=−1,0,1,2,3\displaystyle {x}={-1, 0, 1, 2, 3}.
          Mark the table’s points on the coordinate plane.
          Then show the graph of the function, and check that all your points are on it.

          y=−2x+3\displaystyle y=-2x+3
          Exercise 4

          The lines y=2x+1\displaystyle {y}={2x+1} and y=−x+4\displaystyle {y}={-x+4} meet at one point.
          Mark it.
          Show by substituting that the point is on both lines.

          y=2x+1\displaystyle y=2x+1
          y=−x+4\displaystyle y=-x+4