Question

which table represents a linear function?

table 1:
\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline 1 & \\frac{1}{2} \\\\ \\hline 2 & 1 \\\\ \\hline 3 & 1\\frac{1}{2} \\\\ \\hline 4 & 2 \\\\ \\hline \\end{array}$$

table 2:
\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline 1 & 1 \\\\ \\hline 2 & \\frac{1}{2} \\\\ \\hline 3 & \\frac{1}{3} \\\\ \\hline 4 & \\frac{1}{4} \\\\ \\hline \\end{array}$$

table 3:
\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline 1 & 7 \\\\ \\hline 2 & 9 \\\\ \\hline 3 & 13 \\\\ \\hline 4 & 21 \\\\ \\hline \\end{array}$$

table 4:
\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline 1 & 0 \\\\ \\hline 2 & 6 \\\\ \\hline 3 & 16 \\\\ \\hline 4 & 30 \\\\ \\hline \\end{array}$$

Explanation:

⚡ Using what you learned: understanding linear functions

Step 1: Analyze the definition of a linear function

A table represents a linear function if the rate of change (the change in \(y\) divided by the change in \(x\)) is constant. Since the \(x\)-values in all tables increase by a constant step of \(1\), we only need to find the table where the difference between consecutive \(y\)-values is constant:

$$ \Delta y = y_{n+1} - y_n = \text{constant} $$

Step 2: Test the first table

Check the differences between consecutive \(y\)-values:

• From \(y = \frac{1}{2}\) to \(y = 1\):

$$ 1 - \frac{1}{2} = \frac{1}{2} $$

• From \(y = 1\) to \(y = 1\frac{1}{2}\):

$$ 1\frac{1}{2} - 1 = \frac{1}{2} $$

• From \(y = 1\frac{1}{2}\) to \(y = 2\):

$$ 2 - 1\frac{1}{2} = \frac{1}{2} $$

The difference is constantly \(\frac{1}{2}\).

Step 3: Verify other tables

• Second table: Differences are \(\frac{1}{2} - 1 = -\frac{1}{2}\), then \(\frac{1}{3} - \frac{1}{2} = -\frac{1}{6}\) (not constant).
• Third table: Differences are \(9 - 7 = 2\), then \(13 - 9 = 4\) (not constant).
• Fourth table: Differences are \(6 - 0 = 6\), then \(16 - 6 = 10\) (not constant).

Answer:

The first table represents a linear function:

x	y
2	1
3	\(1\frac{1}{2}\)
4	2