Liam is using sequences to compare the growth rates of h(x)=1.2x

Question

and j(x)=1.2x
. Which statement correctly describes how Liam should do this and what he will observe?(1 point)
Responses

Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of h(x)=1.2x
is greater than the growth rate of j(x)=1.2x
when its terms are greater.
Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of h left parenthesis x right parenthesis equals 1.2 x is greater than the growth rate of j left parenthesis x right parenthesis equals 1.2 superscript x baseline when its terms are greater.

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 j left parenthesis x right parenthesis equals 1.2 superscript x baseline will quickly surpass the growth rate of h left parenthesis x right parenthesis equals 1.2 x .

Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of j(x)=1.2x
is only greater than the growth rate of h(x)=1.2x
when its terms are greater.
Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of j left parenthesis x right parenthesis equals 1.2 superscript x baseline is only greater than the growth rate of h left parenthesis x right parenthesis equals 1.2 x when its terms are greater.

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 h left parenthesis x right parenthesis equals 1.2 x will quickly surpass the growth rate of j left parenthesis x right parenthesis equals 1.2 superscript x baseline .
Skip to navigation

Answer 1

To compare the growth rates of the functions h(x) = 1.2x (which is a linear function) and j(x) = 1.2^x (which is an exponential function), Liam should analyze their behavior as x increases.

The correct statement among the options provided is:

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

This is accurate because, while h(x) increases linearly, j(x) grows exponentially, meaning that as x becomes larger, j(x) will outpace h(x) significantly.

Answer 2

Austin is using graphs to compare the growth rates of g(x)=1.3x
and f(x)=1.3x
. Which statement correctly describes how Austin should do this and what he will observe? (1 point)
Responses

Austin should compare the steepness of the curves. The growth rate of f(x)=1.3x
will quickly surpass the growth rate of g(x)=1.3x
.
Austin should compare the steepness of the curves. The growth rate of f left parenthesis x right parenthesis equals 1.3 superscript x baseline will quickly surpass the growth rate of g left parenthesis x right parenthesis equals 1.3 x .

Austin should compare the steepness of the curves. The growth rate of g(x)=1.3x
will quickly surpass the growth rate of f(x)=1.3x
.
Austin should compare the steepness of the curves. The growth rate of g left parenthesis x right parenthesis equals 1.3 x will quickly surpass the growth rate of f left parenthesis x right parenthesis equals 1.3 superscript x baseline .

Austin should find where one curve is above the other curve on the graph. The growth rate of f(x)=1.3x
is only greater than the growth rate of g(x)=1.3x
to the right of their right-most intersection point.
Austin should find where one curve is above the other curve on the graph. The growth rate of f left parenthesis x right parenthesis equals 1.3 superscript x baseline is only greater than the growth rate of g left parenthesis x right parenthesis equals 1.3 x to the right of their right-most intersection point.

Austin should find where one curve is above the other curve on the graph. The growth rate of g(x)=1.3x
is greater than the growth rate of f(x)=1.3x
between the intersection points of the curve.
Austin should find where one curve is above the other curve on the graph. The growth rate of g left parenthesis x right parenthesis equals 1.3 x is greater than the growth rate of f left parenthesis x right parenthesis equals 1.3 superscript x baseline between the intersection points of the curve.
Skip to navigation

Answer 3

To compare the growth rates of the functions g(x) = 1.3x (which is a linear function) and f(x) = 1.3^x (which is an exponential function), Austin should analyze their graphs and observe how they behave as x increases.

The correct statement among the options provided is:

"Austin should compare the steepness of the curves. The growth rate of f(x) = 1.3^x will quickly surpass the growth rate of g(x) = 1.3x."

This is the accurate interpretation because the exponential function f(x) = 1.3^x grows much faster than the linear function g(x) = 1.3x as x increases, and Austin would observe that the graph of f(x) becomes steeper than that of g(x) for larger values of x.

Answer 4

Use the image to answer the question.

A coordinate plane has an increasing solid curve and a dotted straight line plotted on it, corresponding to two functions.

Given that the graph f(x)
is linear and the graph g(x)
is exponential solve the following problem:

Consider the graph of Function #1 and Function #2. How do the growth rates of the two functions compare?

(1 point)
Responses

The growth rate of Function #1 is always greater than the growth rate of Function #2.
The growth rate of Function #1 is always greater than the growth rate of Function #2.

Initially, Function #2 has the greater growth rate. After x=32
, the growth rate of Function #1 surpasses the growth rate of Function #2.
Initially, Function #2 has the greater growth rate. After x equals 32 , the growth rate of Function #1 surpasses the growth rate of Function #2.

The growth rate of Function #2 is always greater than the growth rate of Function #1.
The growth rate of Function #2 is always greater than the growth rate of Function #1.

Initially, Function #1 has the greater growth rate. After x=32
, the growth rate of Function #2 surpasses the growth rate of Function #1.
Initially, Function #1 has the greater growth rate. After x equals 32 , the growth rate of Function #2 surpasses the growth rate of Function #1.
Skip to navigation

Answer 5

Based on the information provided (Function #1 is linear and Function #2 is exponential), we can infer how their growth rates compare.

The correct response is:

"Initially, Function #2 has the greater growth rate. After x = 32, the growth rate of Function #1 surpasses the growth rate of Function #2."

This assertion aligns with the common behavior of linear versus exponential functions: while the exponential function often has a higher growth rate at the beginning, the linear function will eventually catch up and surpass the exponential function at some point, after which the exponential function will again increase rapidly. In your case, it seems that after x = 32, the linear function's growth is greater than that of the exponential function.

Answer 6

Use the image to answer the question.

A coordinate plane shows an increasing solid curve and a dotted straight line plotted on it, corresponding to two functions.

Marshall is comparing the growth rates of f(x)=6⋅1.5x
and g(x)=5x+10
using this graph. Based on the graph, he concludes that the growth rate of g(x)=5x+10
is always greater than the growth rate of f(x)=6⋅1.5x
. Where is his mistake?

(1 point)
Responses

Marshall has it backward. The growth rate of f(x)=6⋅1.5x
is always greater than the growth rate of g(x)=5x+10
.
Marshall has it backward. The growth rate of f left parenthesis x right parenthesis equals 6 times 1.5 superscript x baseline is always greater than the growth rate of g left parenthesis x right parenthesis equals 5 x plus 10 .

The growth rate of g(x)=5x+10
is only greater than the growth rate of f(x)=6⋅1.5x
when x
is greater than approximately −1.3
.
The growth rate of g left parenthesis x right parenthesis equals 5 x plus 10 is only greater than the growth rate of f left parenthesis x right parenthesis equals 6 times 1.5 superscript x baseline when x is greater than approximately negative 1.3 .

Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f(x)=6⋅1.5x
eventually surpasses the growth rate of g(x)=5x+10
for large values of x
.
Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f left parenthesis x right parenthesis equals 6 times 1.5 superscript x baseline eventually surpasses the growth rate of g left parenthesis x right parenthesis equals 5 x plus 10 for large values of x .

Marshall is only considering a small portion of the domains of the functions. If he went farther to the left on the graph, he would observe that the growth rate of f(x)=6⋅1.5x
eventually surpasses the growth rate of g(x)=5x+10
for negative values of x
.
Marshall is only considering a small portion of the domains of the functions. If he went farther to the left on the graph, he would observe that the growth rate of f left parenthesis x right parenthesis equals 6 times 1.5 superscript x baseline eventually surpasses the growth rate of g left parenthesis x right parenthesis equals 5 x plus 10 for negative values of x .
Skip to navigation

Answer 7

To analyze Marshall's conclusion about the growth rates of f(x) = 6⋅1.5^x and g(x) = 5x + 10, we need to recognize the behavior of these functions:

1. Exponential Growth: The function f(x) = 6⋅1.5^x is an exponential function, which generally grows faster than a linear function as x becomes large.
2. Linear Growth: The function g(x) = 5x + 10 is a linear function that will continue to increase, but at a constant rate.

Given that exponential functions eventually outgrow linear functions, Marshall's conclusion that g(x) is always greater than f(x) is incorrect.

The correct response to indicate where Marshall made his mistake is:

"Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f(x)=6⋅1.5x eventually surpasses the growth rate of g(x)=5x+10 for large values of x."

This accurately reflects the comparison between the two types of functions and highlights that, over a large enough x, the exponential function will dominate the linear function.