Question

Use the table to answer the question.

This table displays the exponential growth of a function over equal intervals.
x
f(x)=a⋅bx

x0
a⋅bx0

x0+Δx
a⋅b(x0+Δx)

x0+2Δx
a⋅b(x0+2Δx)

Ramona wants to use the table to prove something about how exponential functions grow. What does she need to do next and what will she be able to prove?

(1 point)
Responses

1 Ramona needs to add another column to the table and find the growth factors between the consecutive values of f(x)
. She will be able to prove that exponential functions grow by equal factors over equal intervals.

2 Ramona needs to subtract the x -values from the f left parenthesis x right parenthesis -values. She will be able to prove that exponential functions grow by equal factors over equal intervals.

3 Ramona needs to subtract the x -values from the f left parenthesis x right parenthesis -values. She will be able to prove that exponential functions grow by equal differences over equal intervals.

4 Ramona needs to add another column to the table and find the change in f(x)
-values. She will be able to prove that exponential functions grow by equal differences over equal intervals.

Answer 1

Answer: 1.

She should add a column computing the growth factors f(x+Δx)/f(x). For f(x)=a·b^x,
f(x0+Δx)/f(x0) = (a·b^{x0+Δx})/(a·b^{x0}) = b^{Δx},
and similarly for the next interval, so the factor is the same. Thus exponential functions grow by equal factors over equal intervals.