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A function's table of values

Algebra, grade 8 Lesson 2 of 27Learning

A function's table of values

Table of values

A function’s table of values shows some values of x\displaystyle x, and under each one the y\displaystyle y the function gives it.

  • Each column of the table is an ordered pair (x,y)\displaystyle (x,y).
  • The values of x\displaystyle x are chosen; the values of y\displaystyle y are worked out by the rule.
Filling in a table from a rule

The rule: multiply by 2 and subtract 3.
We work out y\displaystyle y for each x\displaystyle x in the table:

  • x=−2\displaystyle {x}={-2}: (−2)⋅2−3=−4−3=−7\displaystyle {(-2)\cdot2-3}={-4-3}={-7}.
  • x=0\displaystyle {x}={0}: 0⋅2−3=−3\displaystyle {0\cdot2-3}={-3}.
  • x=3\displaystyle {x}={3}: 3⋅2−3=6−3=3\displaystyle {3\cdot2-3}={6-3}={3}.

And the same in the other columns:

x\displaystyle x−2\displaystyle -2−1\displaystyle -10\displaystyle 01\displaystyle 12\displaystyle 23\displaystyle 3
y\displaystyle y−7\displaystyle -7−5\displaystyle -5−3\displaystyle -3−1\displaystyle -11\displaystyle 13\displaystyle 3
A table that is not a function’s

A table in which the same x\displaystyle x gets two different values of y\displaystyle y is not a function’s table.

x\displaystyle x1\displaystyle 12\displaystyle 22\displaystyle 23\displaystyle 3
y\displaystyle y4\displaystyle 45\displaystyle 57\displaystyle 76\displaystyle 6

Here x=2\displaystyle {x}={2} gets both 5 and 7.

Finding the rule from a table
Check the rule in every column, not in one column only.

In this table y\displaystyle y grows by 3 each time x\displaystyle x grows by 1:

x\displaystyle x0\displaystyle 01\displaystyle 12\displaystyle 23\displaystyle 34\displaystyle 4
y\displaystyle y2\displaystyle 25\displaystyle 58\displaystyle 811\displaystyle 1114\displaystyle 14
  • A fixed step of 3 in every column means the rule has “multiply by 3” in it.
  • 3⋅x\displaystyle 3\cdot x gives 0, 3, 6, 9, 12. In every column y\displaystyle y is 2 more than that.
  • The rule: multiply by 3 and add 2. A check in the last column: 4⋅3+2=14\displaystyle {4\cdot3+2}={14}.
A squaring rule

In this table y\displaystyle y does not change by the same step: it goes down by 3, then by 1, then up by 1 and by 3.

x\displaystyle x−2\displaystyle -2−1\displaystyle -10\displaystyle 01\displaystyle 12\displaystyle 2
y\displaystyle y4\displaystyle 41\displaystyle 10\displaystyle 01\displaystyle 14\displaystyle 4

The rule here is “square x\displaystyle x”: (−2)2=4\displaystyle {(-2)^2}={4}, (−1)2=1\displaystyle {(-1)^2}={1}, 02=0\displaystyle {0^2}={0}.

Exercise 1

The rule: multiply by −2\displaystyle -2 and add 5.
Fill in the table of values.

x\displaystyle x−2\displaystyle -2−1\displaystyle -10\displaystyle 01\displaystyle 12\displaystyle 23\displaystyle 3
y\displaystyle y
Exercise 2

Find the rule of each table, and use it to work out y\displaystyle y when x=10\displaystyle {x}={10}.

a
x\displaystyle x0\displaystyle 01\displaystyle 12\displaystyle 23\displaystyle 34\displaystyle 4
y\displaystyle y1\displaystyle 15\displaystyle 59\displaystyle 913\displaystyle 1317\displaystyle 17

    / – 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
    x\displaystyle x1\displaystyle 12\displaystyle 23\displaystyle 34\displaystyle 45\displaystyle 5
    y\displaystyle y10\displaystyle 109\displaystyle 98\displaystyle 87\displaystyle 76\displaystyle 6

      / – 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
      x\displaystyle x1\displaystyle 12\displaystyle 23\displaystyle 34\displaystyle 4
      y\displaystyle y2\displaystyle 25\displaystyle 510\displaystyle 1017\displaystyle 17

        / – 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

        Which of the tables is a function’s table? Explain.

        Table 1:

        x\displaystyle x1\displaystyle 12\displaystyle 23\displaystyle 34\displaystyle 4
        y\displaystyle y5\displaystyle 55\displaystyle 55\displaystyle 55\displaystyle 5

        Table 2:

        x\displaystyle x3\displaystyle 33\displaystyle 34\displaystyle 45\displaystyle 5
        y\displaystyle y1\displaystyle 12\displaystyle 23\displaystyle 34\displaystyle 4

        Table 3:

        x\displaystyle x−1\displaystyle -10\displaystyle 01\displaystyle 12\displaystyle 2
        y\displaystyle y1\displaystyle 10\displaystyle 01\displaystyle 14\displaystyle 4
        Exercise 4

        At a swimming pool, a membership card costs 20 shekels.
        Each visit costs 15 shekels more.

        • Make a table of values: x\displaystyle x – the number of visits, from 1 to 5; y\displaystyle y – how much you paid in all.
        • Dana paid 155 shekels in all. How many times did she go to the pool? Write an equation and solve it.