Linear vs. Exponential Growth Rates Quick Check

Question

5 of 55 of 5 Items

Question
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 .

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 .

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 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 1

Marshall's mistake lies in not considering the long-term behavior of the two functions. The correct response 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⋅1.5^x eventually surpasses the growth rate of g(x)=5x+10 for large values of x.

This is because exponential functions (like f(x)=6⋅1.5^x) eventually grow faster than linear functions (like g(x)=5x+10) as x increases.