Question

a scientist is conducting an experiment on two types of bacteria to determine which type will grow faster in a pool. after collecting data for two weeks, she finds their growth rates as follows: bacteria 1: $y = 4^x$ bacteria 2: $y = 4x^2$ in these models, $y$ represents the number of bacteria colonies and $x$ represents the number of hours. how do you know which bacteria is growing faster? a quadratic function grows faster than an exponential function. an exponential function grows faster than a quadratic function. an exponential function grows faster than a linear function.

Explanation:

Step1: Analyze the functions

Bacteria 1: \( y = 4^x \) (exponential function, base \( 4>1 \))
Bacteria 2: \( y = 4x \) (linear function, slope \( 4 \))

Step2: Compare growth rates

For exponential function \( y = a^x \) (\( a>1 \)), it grows faster than linear (and quadratic for large \( x \)) as \( x \) increases. Let's test \( x = 10 \) (2 weeks = 14 days, but any large \( x \) works):

• Bacteria 1: \( y = 4^{10}=1048576 \)
• Bacteria 2: \( y = 4\times10 = 40 \)

Exponential (\( y = 4^x \)) grows faster than linear (\( y = 4x \)). Also, exponential grows faster than quadratic for large \( x \) (since exponential has unbounded growth rate, quadratic's rate is linear).

Step3: Match the statement

The correct statement is "An exponential function grows faster than a quadratic function" (wait, no—wait, Bacteria 2 is linear? Wait, no, the options: Wait, the options are:

1. A quadratic function grows faster than an exponential function. (False, exponential grows faster for large \( x \))
2. An exponential function grows faster than a quadratic function. (True, as exponential \( a^x \) ( \( a>1 \)) outpaces quadratic \( ax^2 \) for large \( x \))
3. An exponential function grows faster than a linear function. (Also true, but let's check the models: Bacteria 1 is \( 4^x \) (exponential), Bacteria 2 is \( 4x \) (linear). Wait, maybe the options were misread. Wait, the problem's options:

Looking at the image:

Option 1: "A quadratic function grows faster than an exponential function." (False)

Option 2: "An exponential function grows faster than a quadratic function." (True, because exponential growth (like \( 4^x \)) has a rate that increases, while quadratic ( \( ax^2 \)) has a linear rate of increase. For large \( x \), exponential dominates.)

Option 3: "An exponential function grows faster than a linear function." (Also true, but let's check the models. Bacteria 1: \( y = 4^x \) (exponential), Bacteria 2: \( y = 4x \) (linear). So \( 4^x \) grows faster than \( 4x \). But the options: the middle option is "An exponential function grows faster than a quadratic function"—wait, maybe the Bacteria 2 was a quadratic? Wait, no, the model for Bacteria 2 is \( y = 4x \)? Wait, the image says "Bacteria 2: \( y = 4x \)"? Wait, no, maybe a typo. Wait, the user's image: "Bacteria 1: \( y = 4^x \), Bacteria 2: \( y = 4x \)"? Or maybe Bacteria 2 is \( y = 4x^2 \) (quadratic)? Wait, the options include "quadratic" and "exponential". Let's re-express:

The key is: Exponential functions (with base >1) grow faster than quadratic functions for sufficiently large \( x \), because the derivative of \( a^x \) is \( \ln(a)a^x \) (grows exponentially), while derivative of \( ax^2 \) is \( 2ax \) (grows linearly). So exponential's growth rate outpaces quadratic's.

So the correct option is the one stating "An exponential function grows faster than a quadratic function."

Answer:

An exponential function grows faster than a quadratic function.