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 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.

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
.

Answer 1

It appears there is a misunderstanding in the function definitions you've provided. The functions you wrote are g(x) = 1.3x (a linear function) and f(x) = 1.3^x (an exponential function).

To correctly describe how Austin should analyze the growth rates of these functions, the relevant aspects are:

1. **Finding where one curve is above the other**: Since f(x) = 1.3^x grows exponentially, while g(x) = 1.3x grows linearly, the exponential function will eventually surpass the linear function for larger values of x.

2. **Observing growth behaviors**: The growth rate of the exponential function (f(x)) is only greater than that of the linear function (g(x)) as x increases, especially after their intersection point.

Given this information, the appropriate response would be:

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

In your provided options, this corresponds to the second version (with formatting) of the response about intersection points.