Question

which table of values represents a linear function?

a

x	y
-1	-2
3	1
7	4

b

x	y
2	-1
4	1
6	3

c

x	y
-4	3
1	1
6	-1

d

x	y
3	1
5	-1
7	-3

Explanation:

Step1: Check linearity (constant slope)

A linear function has a constant rate of change (slope), calculated as $m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$ for consecutive points.

Step2: Analyze Table A

$\Delta x$ between points: $-1-(-5)=4$, $3-(-1)=4$, $7-3=4$
$\Delta y$ between points: $-2-(-6)=4$, $1-(-2)=3$, $4-1=3$
Slopes: $\frac{4}{4}=1$, $\frac{3}{4}$, $\frac{3}{4}$ → Not constant.

Step3: Analyze Table B

$\Delta x$ between points: $2-0=2$, $4-2=2$, $6-4=2$
$\Delta y$ between points: $-1-(-3)=2$, $1-(-1)=2$, $3-1=2$
Slopes: $\frac{2}{2}=1$, $\frac{2}{2}=1$, $\frac{2}{2}=1$ → Constant.

Step4: Verify Table C (optional)

$\Delta x$ between points: $-4-(-9)=5$, $1-(-4)=5$, $6-1=5$
$\Delta y$ between points: $3-6=-3$, $1-3=-2$, $-1-1=-2$
Slopes: $\frac{-3}{5}$, $\frac{-2}{5}$, $\frac{-2}{5}$ → Not constant.

Step5: Verify Table D (optional)

$\Delta x$ between points: $3-0=3$, $5-3=2$, $7-5=2$
$\Delta y$ between points: $1-3=-2$, $-1-1=-2$, $-3-(-1)=-2$
Slopes: $\frac{-2}{3}$, $\frac{-2}{2}=-1$, $\frac{-2}{2}=-1$ → Not constant.

Answer:

B.

$x$	$y$
2	-1
4	1
6	3