Question

look at this table:

x	y
-2	-1
-1	-2
0	-4
1	-8
2	-16

write a linear ($y = mx + b$), quadratic ($y = ax^2$), or exponential ($y = a(b)^x$) function that models the data.
$y = \square$

Explanation:

Step1: Check linear fit (constant slope)

Slope between $(-2,-1)$ & $(-1,-2)$: $\frac{-2 - (-1)}{-1 - (-2)} = \frac{-1}{1} = -1$
Slope between $(-1,-2)$ & $(0,-4)$: $\frac{-4 - (-2)}{0 - (-1)} = \frac{-2}{1} = -2$
Slopes are not equal, so not linear.

Step2: Check quadratic fit (constant 2nd difference)

1st differences: $-2-(-1)=-1$, $-4-(-2)=-2$, $-8-(-4)=-4$, $-16-(-8)=-8$
2nd differences: $-2-(-1)=-1$, $-4-(-2)=-2$, $-8-(-4)=-4$
2nd differences are not constant, so not quadratic.

Step3: Check exponential fit (constant ratio)

Ratio of consecutive $y$-values: $\frac{-2}{-1}=2$, $\frac{-4}{-2}=2$, $\frac{-8}{-4}=2$, $\frac{-16}{-8}=2$
Common ratio $b=2$. Use $x=0, y=-4$ to find $a$:
$-4 = a(2)^0 \implies a=-4$

Answer:

$y = -4(2)^x$