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Which of the following tables represents a linear function?

x −2 −1 0 1 2
y 4 1 −2 −5 −8
x −2 −1 0 1 2
y 4 1 0 1 4
x 2 2 0 2 2
y −2 −1 0 1 2
x 0 1 2 3 4
y −2 1 0 1 −2

1 answer

To determine which of the given tables represents a linear function, we need to check if the ratio of change in \( y \) over the change in \( x \) (the slope) is constant for each set of \( (x, y) \) values.

Table 1: \[ \begin{array}{|c|c|} \hline x & y \ \hline -2 & 4 \ -1 & 1 \ 0 & -2 \ 1 & -5 \ 2 & -8 \ \hline \end{array} \]

Calculate the differences:

  • From \(-2\) to \(-1\): \(y\) changes from \(4\) to \(1\) (change = \(-3\))
  • From \(-1\) to \(0\): \(y\) changes from \(1\) to \(-2\) (change = \(-3\))
  • From \(0\) to \(1\): \(y\) changes from \(-2\) to \(-5\) (change = \(-3\))
  • From \(1\) to \(2\): \(y\) changes from \(-5\) to \(-8\) (change = \(-3\))

All \(y\) changes are \(-3\) for a consistent \(x\) increase of \(1\). Thus, this represents a linear function.

Table 2: \[ \begin{array}{|c|c|} \hline x & y \ \hline -2 & 4 \ -1 & 1 \ 0 & 0 \ 1 & 1 \ 2 & 4 \ \hline \end{array} \]

Calculate the differences:

  • From \(-2\) to \(-1\): \(y\) changes from \(4\) to \(1\) (change = \(-3\))
  • From \(-1\) to \(0\): \(y\) changes from \(1\) to \(0\) (change = \(-1\))
  • From \(0\) to \(1\): \(y\) changes from \(0\) to \(1\) (change = \(1\))
  • From \(1\) to \(2\): \(y\) changes from \(1\) to \(4\) (change = \(3\))

Changes are not consistent. Thus, this is not a linear function.

Table 3: \[ \begin{array}{|c|c|} \hline x & y \ \hline 2 & -2 \ 2 & -1 \ 0 & 0 \ 2 & 1 \ 2 & 2 \ \hline \end{array} \]

Notice that there is a constant \(x = 2\) with multiple \(y\) values. Since for one \(x\) value there are multiple corresponding \(y\) values, this does not represent a function, hence cannot be linear.

Table 4: \[ \begin{array}{|c|c|} \hline x & y \ \hline 0 & -2 \ 1 & 1 \ 2 & 0 \ 3 & 1 \ 4 & -2 \ \hline \end{array} \]

Calculate the differences:

  • From \(0\) to \(1\): \(y\) changes from \(-2\) to \(1\) (change = \(3\))
  • From \(1\) to \(2\): \(y\) changes from \(1\) to \(0\) (change = \(-1\))
  • From \(2\) to \(3\): \(y\) changes from \(0\) to \(1\) (change = \(1\))
  • From \(3\) to \(4\): \(y\) changes from \(1\) to \(-2\) (change = \(-3\))

The changes are not consistent, so this is not a linear function.

Conclusion: The only table that represents a linear function is Table 1.