which table represents a linear function?\n| x | y |\n|----|----|\n| 1 | 3 |\n| 2 | 6 |\n| 3 | 12 |\n| 4 |…

Explanation:

Step1: Recall linear - function property

A linear function has a constant rate of change (slope). The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Check the first table

For the first table with points $(1,3),(2,6),(3,12),(4,24)$: $m_1=\frac{6 - 3}{2 - 1}=3$, $m_2=\frac{12 - 6}{3 - 2}=6$, $m_1\neq m_2$, not a linear function.

Step3: Check the second table

For the second table with points $(1,2),(2,5),(3,9),(4,14)$: $m_1=\frac{5 - 2}{2 - 1}=3$, $m_2=\frac{9 - 5}{3 - 2}=4$, $m_1\neq m_2$, not a linear function.

Step4: Check the third table

For the third table with points $(1,-3),(2,-5),(3,-7),(4,-9)$: $m_1=\frac{-5+3}{2 - 1}=-2$, $m_2=\frac{-7 + 5}{3 - 2}=-2$, $m_3=\frac{-9+7}{4 - 3}=-2$. The slope is constant, so it is a linear function.

Step5: Check the fourth table

For the fourth table with points $(1,-2),(2,-4),(3,-2),(4,0)$: $m_1=\frac{-4 + 2}{2 - 1}=-2$, $m_2=\frac{-2+4}{3 - 2}=2$, $m_1\neq m_2$, not a linear function.

Answer:

The table with $x$ - values $1,2,3,4$ and $y$ - values $-3,-5,-7,-9$ represents a linear function.