Use the image to answer the question.

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

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.

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 #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.

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.
Skip to navigation

Answer 1

Based on the characteristics of linear and exponential functions:

1. A linear function, like Function #1 (the solid curve), grows at a constant rate. This means that the rate of change (slope) remains the same as \( x \) increases.

2. An exponential function, like Function #2 (the dotted line), grows at an increasing rate. This implies that as \( x \) increases, the rate of growth of the function accelerates.

Given your description that after \( x = 32 \), the growth rate of Function #2 surpasses that of Function #1, we can deduce that:

• Initially, Function #1 (linear) has the greater growth rate because it starts with a fixed slope.
• After some point, specifically after \( x = 32 \), Function #2 (exponential) grows increasingly faster than Function #1.

Thus, the correct response is:

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