Use the table to answer the question.

Question

x f(x)=a⋅bx
x0 a⋅𝑏𝑥^0
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?

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 factors 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 add another column to the table and find the change in f(x)-values. She will be able to

Answer 1

The best response for Ramona's next step 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.

By calculating the growth factors, she can demonstrate that the increase in the function value (f(x)) over equal intervals of x is consistent, which is characteristic of exponential functions.

Answer 2

Liam is using sequences to compare the growth rates of h(x)=1.2x and j(x)=1.2x . Which statement correctly describes how Liam should do this and what he will observe?
Responses

Liam should compare the rates of change of the terms in both sequences. The growth rate of h(x)=1.2x will quickly surpass the growth rate of j(x)=1.2x

Liam should compare the rates of change of the terms in both sequences. The growth rate of j(x)=1.2x will quickly surpass the growth rate of h(x)=1.2x

Liam should compare the rates of change of the terms in both sequences. The growth rate of h (x)= 1.2x is greater then the growth of J (x) =1.2^x when its terms are greater. Liam Should look at where one sequence has greater terms then in the other sequence. The growth rate of h (x)=1.2x when its terms are greater.

Answer 3

The best response for Liam's situation is:

Liam should compare the rates of change of the terms in both sequences. The growth rate of h(x) = 1.2x is greater than the growth of j(x) = 1.2^x when its terms are greater.

This response accurately reflects the nature of the functions involved, as \( h(x) = 1.2x \) is linear growth while \( j(x) = 1.2^x \) is exponential growth. While \( h(x) \) increases at a constant rate, \( j(x) \) increases at an increasing rate, and thus it will surpass \( h(x) \) as \( x \) becomes sufficiently large.