Linear vs. Exponential Growth Rates Quick Check

Question

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Question
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 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 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 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.
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Answer 1

To correctly analyze the growth rates of the functions \( g(x) = 1.3x \) (linear growth) and \( f(x) = 1.3^x \) (exponential growth), Austin should compare their growth behaviors.

The correct statement 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 \( g(x) = 1.3x \).

Exponential functions grow significantly faster than linear functions as \( x \) increases. Thus, after a certain point, the graph of \( f(x) = 1.3^x \) will rise steeply above that of \( g(x) = 1.3x \).