Question

which table of values represents a linear function?
a

x	y
5	3
7	1
9	0

b

x	y
4	2
6	1
8	0

c

x	y
4	3
5	-3
6	-9

d

x	y
1	1
3	-3
5	-6

Explanation:

Step1: Define linear function test

A linear function has a constant rate of change, meaning $\frac{\Delta y}{\Delta x}$ is the same for all consecutive pairs of points.

Step2: Check Table A

Calculate $\Delta x$ and $\Delta y$:

• $\Delta x_1=5-3=2$, $\Delta y_1=3-5=-2$, rate: $\frac{-2}{2}=-1$
• $\Delta x_2=7-5=2$, $\Delta y_2=1-3=-2$, rate: $\frac{-2}{2}=-1$
• $\Delta x_3=9-7=2$, $\Delta y_3=0-1=-1$, rate: $\frac{-1}{2}=-0.5$

Rates are not equal.

Step3: Check Table B

Calculate $\Delta x$ and $\Delta y$:

• $\Delta x_1=4-1=3$, $\Delta y_1=2-3=-1$, rate: $\frac{-1}{3}$
• $\Delta x_2=6-4=2$, $\Delta y_2=1-2=-1$, rate: $\frac{-1}{2}$

Rates are not equal.

Step4: Check Table C

Calculate $\Delta x$ and $\Delta y$:

• $\Delta x_1=4-3=1$, $\Delta y_1=3-9=-6$, rate: $\frac{-6}{1}=-6$
• $\Delta x_2=5-4=1$, $\Delta y_2=-3-3=-6$, rate: $\frac{-6}{1}=-6$
• $\Delta x_3=6-5=1$, $\Delta y_3=-9-(-3)=-6$, rate: $\frac{-6}{1}=-6$

Rates are constant.

Step5: Verify Table D (optional)

Calculate $\Delta x$ and $\Delta y$:

• $\Delta x_1=1-(-1)=2$, $\Delta y_1=1-5=-4$, rate: $\frac{-4}{2}=-2$
• $\Delta x_2=3-1=2$, $\Delta y_2=-3-1=-4$, rate: $\frac{-4}{2}=-2$
• $\Delta x_3=5-3=2$, $\Delta y_3=-6-(-3)=-3$, rate: $\frac{-3}{2}=-1.5$

Rates are not equal.

Answer:

C.

$x$	$y$
4	3
5	-3
6	-9