Question

what type of function is represented by the table below and for what reason?

$x$

$y$

-----

-----

0

2.5

2

40

4

640

6

10,240

a linear, because the $x$-values increase at a constant rateb linear, because the graph of the points makes a linec exponential, because the $y$-values for successive integer values of $x$ have a common ratio of 4d exponential, because the $y$-values for successive integer values of $x$ have a common ratio of 16

Explanation:

Step1: Check linearity (constant y-change)

Calculate differences between consecutive y-values:
$40 - 2.5 = 37.5$
$640 - 40 = 600$
$10240 - 640 = 9600$
Differences are not constant, so not linear.

Step2: Check exponential (constant ratio)

Calculate ratios of consecutive y-values:
$\frac{40}{2.5} = 16$
$\frac{640}{40} = 16$
$\frac{10240}{640} = 16$
Ratios are constant (16), so exponential.

Answer:

D. Exponential, because the y-values for successive integer values of x have a common ratio of 16