Question

unit 2
linear & exponential functions → lesson 9
linear vs. exponential growth rates ▶ back to intro page

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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)
○ initially, function #2 has the greater growth rate. after ( x = 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.
○ initially, function #1 has the greater growth rate. after ( x = 32 ), the growth rate of function #2 surpasses the growth rate of function #1.
○ the growth rate of function #1 is always greater than the growth rate of function #2.

Explanation:

• Linear Function (Function #2): A linear function has a constant slope (growth rate), meaning its rate of change doesn’t vary with \( x \). From the graph, it starts with a steeper slope initially.
• Exponential Function (Function #1): An exponential function has a growth rate that increases with \( x \) (its slope becomes steeper as \( x \) increases).
• Key Comparison: Initially, the linear function (Function #2) has a greater growth rate (steeper slope). However, as \( x \) increases (specifically after \( x = 32 \) from the graph’s behavior), the exponential function’s (Function #1) growth rate (slope) becomes steeper than the linear function’s constant slope, so it surpasses Function #2’s growth rate.

Answer:

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