Which Statement Best Describes The Growth Rates Of The Functions Below?1. { Y = 4x^2 $}$2. { Y = 4^x $}$[ \begin{tabular}{|c|c|} \hline x & Y \ \hline 0 & 0 \ \hline 1 & 4 \ \hline 2 & 16 \ \hline 3 & 36 \ \hline 4 & 64

Introduction

In mathematics, the growth rate of a function refers to how quickly the function's output changes as the input increases. This concept is crucial in various fields, including economics, physics, and computer science. In this article, we will explore the growth rates of two functions: $y = 4x^2$ and $y = 4^x$. We will analyze their growth rates using a table of values and discuss which statement best describes their growth rates.

Function 1: $y = 4x^2$

The first function is a quadratic function, which is a polynomial of degree two. The general form of a quadratic function is $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this case, the function is $y = 4x^2$, where $a = 4$ and $b = c = 0$.

To understand the growth rate of this function, let's examine its table of values:

As we can see, the function's output increases rapidly as the input increases. However, the rate of increase is not constant. For example, when $x$ increases from 1 to 2, the output increases from 4 to 16, which is a fourfold increase. When $x$ increases from 2 to 3, the output increases from 16 to 36, which is a two-and-a-half times increase. This indicates that the function's growth rate is not constant, but rather it accelerates as the input increases.

Function 2: $y = 4^x$

The second function is an exponential function, which is a function of the form $y = ab^x$, where $a$ and $b$ are constants. In this case, the function is $y = 4^x$, where $a = 1$ and $b = 4$.

To understand the growth rate of this function, let's examine its table of values:

As we can see, the function's output increases exponentially as the input increases. For example, when $x$ increases from 1 to 2, the output increases from 4 to 16, which is a fourfold increase. When $x$ increases from 2 to 3, the output increases from 16 to 64, which is a fourfold increase. This indicates that the function's growth rate is constant, and it increases by a factor of 4 for every one-unit increase in the input.

Comparing the Growth Rates

Now that we have analyzed the growth rates of both functions, let's compare them. The first function, $y = 4x^2$, has a growth rate that accelerates as the input increases. The second function, $y = 4^x$, has a growth rate that is constant and increases by a factor of 4 for every one-unit increase in the input.

Which statement best describes the growth rates of these functions?

• The growth rate of $y = 4x^2$ is constant and increases by a factor of 4 for every one-unit increase in the input.
• The growth rate of $y = 4x^2$ accelerates as the input increases.
• The growth rate of $y = 4^x$ is constant and increases by a factor of 4 for every one-unit increase in the input.
• The growth rate of $y = 4^x$ accelerates as the input increases.

The correct answer is:

• The growth rate of $y = 4x^2$ accelerates as the input increases.
• The growth rate of $y = 4^x$ is constant and increases by a factor of 4 for every one-unit increase in the input.

Conclusion

In conclusion, the growth rate of a function refers to how quickly the function's output changes as the input increases. The growth rate of a function can be constant or accelerating, depending on the type of function. In this article, we analyzed the growth rates of two functions: $y = 4x^2$ and $y = 4^x$. We found that the growth rate of $y = 4x^2$ accelerates as the input increases, while the growth rate of $y = 4^x$ is constant and increases by a factor of 4 for every one-unit increase in the input.

References

• [1] Khan Academy. (n.d.). Quadratic Functions. Retrieved from https://www.khanacademy.org/math/algebra/quadratic-equations
• [2] Khan Academy. (n.d.). Exponential Functions. Retrieved from https://www.khanacademy.org/math/algebra/exponential-functions

Discussion

Introduction

In our previous article, we explored the growth rates of two functions: $y = 4x^2$ and $y = 4^x$. We analyzed their growth rates using a table of values and discussed which statement best describes their growth rates. In this article, we will answer some frequently asked questions about the growth rates of functions.

Q: What is the growth rate of a function?

A: The growth rate of a function refers to how quickly the function's output changes as the input increases. It is a measure of how fast the function grows or decays.

Q: What are the different types of growth rates?

A: There are two main types of growth rates: constant and accelerating. A constant growth rate means that the function's output increases by a fixed amount for every one-unit increase in the input. An accelerating growth rate means that the function's output increases by a larger amount for every one-unit increase in the input.

Q: How do I determine the growth rate of a function?

A: To determine the growth rate of a function, you can use a table of values or a graph. By examining the table of values or the graph, you can see how quickly the function's output changes as the input increases.

Q: What is the difference between a quadratic function and an exponential function?

A: A quadratic function is a polynomial of degree two, while an exponential function is a function of the form $y = ab^x$. Quadratic functions have a constant growth rate, while exponential functions have an accelerating growth rate.

Q: Can you give an example of a function with a constant growth rate?

A: Yes, the function $y = 2x$ has a constant growth rate. For every one-unit increase in the input, the output increases by 2.

Q: Can you give an example of a function with an accelerating growth rate?

A: Yes, the function $y = 2^x$ has an accelerating growth rate. For every one-unit increase in the input, the output increases by a factor of 2.

Q: How do I use the growth rate of a function in real-world applications?

A: The growth rate of a function can be used in various real-world applications, such as:

• Modeling population growth
• Predicting the spread of diseases
• Analyzing the growth of economies
• Understanding the behavior of complex systems

Q: What are some common mistakes to avoid when analyzing the growth rate of a function?

A: Some common mistakes to avoid when analyzing the growth rate of a function include:

• Assuming a constant growth rate when the function is actually accelerating
• Failing to account for the input's effect on the output
• Not considering the function's domain and range

Conclusion

In conclusion, understanding the growth rate of a function is crucial in various fields, including mathematics, economics, and computer science. By analyzing the growth rate of a function, you can gain insights into its behavior and make predictions about its future performance. We hope this Q&A article has helped you understand the growth rates of functions better.

References

• [1] Khan Academy. (n.d.). Quadratic Functions. Retrieved from https://www.khanacademy.org/math/algebra/quadratic-equations
• [2] Khan Academy. (n.d.). Exponential Functions. Retrieved from https://www.khanacademy.org/math/algebra/exponential-functions

Discussion

Do you have any questions about the growth rates of functions? Share your thoughts and ideas in the comments below!