Question

introduction to linear functions
pre-test

option 1:
\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline 1 & 5 \\\\ \\hline 2 & 9 \\\\ \\hline 3 & 5 \\\\ \\hline 4 & 9 \\\\ \\hline \\end{array}$$

option 2:
\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline 1 & -5 \\\\ \\hline 2 & 10 \\\\ \\hline 3 & -15 \\\\ \\hline 4 & 20 \\\\ \\hline \\end{array}$$

option 3:
\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline 1 & 5 \\\\ \\hline 2 & 10 \\\\ \\hline 3 & 20 \\\\ \\hline 4 & 40 \\\\ \\hline \\end{array}$$

option 4:
\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline 1 & -5 \\\\ \\hline 2 & 0 \\\\ \\hline 3 & 5 \\\\ \\hline 4 & 10 \\\\ \\hline \\end{array}$$

Explanation:

⚡ Using what you learned: understanding linear functions

Step 1: Identify the definition of a linear function

A table represents a linear function if the rate of change (slope) between consecutive points is constant. This means as \( x \) increases by a constant amount, \( y \) must also change by a constant amount.

In all four tables, \( x \) increases by \( 1 \) at each step:

$$ \Delta x = 1 $$

We need to find the table where the change in \( y \) (\( \Delta y \)) is constant.

Step 2: Analyze the tables

• Table 1:
• From \( y = 5 \) to \( 9 \): \( \Delta y = +4 \)
• From \( y = 9 \) to \( 5 \): \( \Delta y = -4 \)
• Not constant.

• Table 2:
• From \( y = -5 \) to \( 10 \): \( \Delta y = +15 \)
• From \( y = 10 \) to \( -15 \): \( \Delta y = -25 \)
• Not constant.

• Table 3:
• From \( y = 5 \) to \( 10 \): \( \Delta y = +5 \)
• From \( y = 10 \) to \( 20 \): \( \Delta y = +10 \)
• Not constant.

• Table 4:
• From \( y = -5 \) to \( 0 \): \( \Delta y = +5 \)
• From \( y = 0 \) to \( 5 \): \( \Delta y = +5 \)
• From \( y = 5 \) to \( 10 \): \( \Delta y = +5 \)
• Constant rate of change: \( m = 5 \).

Answer:

The fourth table represents a linear function:

\( x \)	\( y \)
2	0
3	5
4	10