https://file.aiquickdraw.com/mathbot/file/1729575202463je7llitm.jpg

The question asks which statement best describes the growth rates of two functions: \( y = 4x^2 \) (a quadratic function) and \( y = 4^x \) (an exponential function). The tables provided give the values for \( x \) ranging from 0 to 4.

Let's analyze the growth rates based on the provided values:

- For the quadratic function \( y = 4x^2 \):
  - At \( x = 0 \), \( y = 0 \)
  - At \( x = 1 \), \( y = 4 \)
  - At \( x = 2 \), \( y = 16 \)
  - At \( x = 3 \), \( y = 36 \)
  - At \( x = 4 \), \( y = 64 \)

- For the exponential function \( y = 4^x \):
  - At \( x = 0 \), \( y = 1 \)
  - At \( x = 1 \), \( y = 4 \)
  - At \( x = 2 \), \( y = 16 \)
  - At \( x = 3 \), \( y = 64 \)
  - At \( x = 4 \), \( y = 256 \)

### Growth Comparison:
- **For \( 0 \leq x \leq 2 \)**: Both functions appear to grow at similar rates.
- **For \( x = 3 \)**: The quadratic function gives \( y = 36 \), while the exponential function gives \( y = 64 \), showing that the exponential function grows faster beyond \( x = 2 \).
- **For \( x = 4 \)**: The quadratic function gives \( y = 64 \), while the exponential function gives \( y = 256 \), confirming that the exponential function grows significantly faster from \( x = 3 \) onward.

### Conclusion:
The exponential function grows faster than the quadratic function starting from \( x = 3 \), and this difference continues to widen for larger values of \( x \).

### Correct Answer:
The statement that best describes the growth is:
- "The exponential function grows faster than the quadratic function over two intervals; \( 2 \leq x \leq 4 \)."

Would you like more details or have any other questions?

Here are 5 related questions to consider:
1. How do exponential functions generally compare to polynomial functions in terms of growth?
2. Can we generalize the behavior of exponential functions as \( x \to \infty \)?
3. How would changing the base of the exponential function affect its growth rate?
4. What happens if we increase the coefficient of the quadratic function in this comparison?
5. Could we determine the exact crossover point where the exponential starts to outgrow the quadratic?

**Tip:** Exponential functions eventually outgrow any polynomial function, no matter the degree, as \( x \) increases.

Which statement best describes the growth rates of the functions y = 4x^2 and y = 4^x?

This problem compares the growth rates of the quadratic function y = 4x^2 and the exponential function y = 4^x. Detailed table values for x from 0 to 4 show that the exponential function eventually grows faster beyond x = 2. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation