Use the table to answer the question.

Question

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

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

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

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

Ramona needs to subtract the x
-values from the f(x)
-values. She will be able to prove that exponential functions grow by equal factors over equal intervals.
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.
Skip to navigation

Answer 1

To prove properties of exponential functions, Ramona should focus on how the function behaves over equal intervals. Given that the values of \( f(x) \) are produced by the exponential formula \( f(x) = a \cdot b^x \), the growth of the function can be analyzed by looking at how the function value changes as \( x \) increases by equal steps.

The most applicable step for her would be to observe how each \( f(x) \) value relates to the previous one by calculating the ratio of consecutive \( f(x) \) values. This is referred to as finding growth factors, which are consistent in exponential functions.

Thus, the best response is:

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.