Question

How can we tell whether it's quadratic or exponential from given table below here?

Answer 1

Clearly, it's not A nor D.

Answer 2

but how to know which B or C?

Answer 3

I want your pony mustache

Answer 4

uh oh ._.

Answer 5

exponential functions have the following property: \[
\frac{f(x+\epsilon)}{f(x)} = \frac{f(\epsilon)}{f(0)}
\]

Answer 6

haha what's with uh-oh? @UsukiDoll

and what's \(\epsilon\)? just some constant?

Answer 7

\[
\frac{ce^{a(x+\epsilon)}}{ce^{ax}} = \frac{ce^{ax}e^{a\epsilon}}{ce^{ax}} = e^{a\epsilon }
\]

Answer 8

In this case \(f(x) = ce^{ax}\) to be your general exponential function.

Answer 9

Yeah, \(\epsilon\) is any real number. It's a constant.

Answer 10

In our case, we would let \(\epsilon =1\).

Answer 11

can you show me how to apply that to determine whether it's exponential or not?

Answer 12

So \[
\frac{f(2)}{f(1)} = \frac{f(3)}{f(2)}=\frac{f(4)}{f(3)}
\]

Answer 13

ah I see. that's make sense. Thank you so much!

Answer 14

gimme mustache

Answer 15

Here you go:

Answer 16

For completion\[
\frac{ce^{a(x+\epsilon)}}{ce^{ax}} = \frac{ce^{ax}e^{a\epsilon}}{ce^{ax}} = e^{a\epsilon } = \frac{ce^{a\epsilon}}{ce^{a\cdot 0}}
\]In this case \(f(x) = ce^{ax}\).