Question 1 A) Which function has the greater estimated average rate of change over the interval [0,1], the exponential function y=4xy=4xor the quadratic function y=4x2y=4x2? (1 point) is the function with the greater estimated average rate of change on the interval. Question 2 A) Use the following image to answer the question. A quadratic function is graphed. Compare the estimated average rate of change for the quadratic function to the estimated average rate of change for the exponential function y=6xy=6x on the interval [0,5]. Use one of the symbols <,=, or > to describe the comparison (1 point) The estimated average rate of change of the quadratic function the estimated average rate of change of the exponential function. Question 3 A)Compare the estimated average rate of change of the exponential functioin y=9(13)xy=9(13)xand the quadratic function y=9x2+3x. y=9x2+3x. Which function has a negative estimated average rate of change over the interval [1,3]?(1 point) Responses the exponential function the exponential function Neither function Neither function the quadratic function the quadratic function Both functions Both functions Question 4 A) Use the image to answer the question Compare the estimate average rate of change of the graphed quadratic function over the interval [2,6] to the average rate of change of the exponential function y=34(2)xy=34(2)xover the same interval. (1 point) Responses The estimated average rate of change of the quadratic function is 0.16 greater than the estimated average rate of change of the exponential function of the interval [2,6]. The estimated average rate of change of the quadratic function is 0.16 greater than the estimated average rate of change of the exponential function of the interval [2,6]. The estimated average rate of change of the exponential function is 7.25 greater than the estimated average rate of change of the Quadratic function of the interval [2,6]. The estimated average rate of change of the exponential function is 7.25 greater than the estimated average rate of change of the Quadratic function of the interval [2,6]. The estimated average rate of change of the exponential function is 0.16 greater than the estimated average rate of change of the Quadratic function of the interval [2,6]. The estimated average rate of change of the exponential function is 0.16 greater than the estimated average rate of change of the Quadratic function of the interval [2,6]. The estimated average rate of change of the quadratic function is 7.25 greater than the estimated average rate of change of the exponential function of the interval [2,6]. The estimated average rate of change of the quadratic function is 7.25 greater than the estimated average rate of change of the exponential function of the interval [2,6].

1 answer

To answer these questions, we will calculate the estimated average rates of change for the functions given in the intervals specified.

Question 1

A) For the exponential function \( y = 4^x \) and the quadratic function \( y = 4x^2 \) over the interval \([0,1]\):

  • Exponential function \( y = 4^x \):

    • At \( x = 0 \): \( y(0) = 4^0 = 1 \)
    • At \( x = 1 \): \( y(1) = 4^1 = 4 \)
    • Average rate of change = \( \frac{y(1) - y(0)}{1 - 0} = \frac{4 - 1}{1 - 0} = 3 \)
  • Quadratic function \( y = 4x^2 \):

    • At \( x = 0 \): \( y(0) = 4(0^2) = 0 \)
    • At \( x = 1 \): \( y(1) = 4(1^2) = 4 \)
    • Average rate of change = \( \frac{y(1) - y(0)}{1 - 0} = \frac{4 - 0}{1 - 0} = 4 \)

Conclusion: The quadratic function \( y = 4x^2 \) has a greater estimated average rate of change.

Question 2

A) Compare estimated average rates of change for the quadratic function and the exponential function \( y = 6^x \) over the interval \([0,5]\).

Since we do not have the specific quadratic function or image to derive its values, a general assumption:

  • Typically, the exponential function grows more rapidly than quadratic functions for larger intervals.
  • Therefore, for \( x = 0 \) to \( x = 5 \):

Conclusion: The average rate of change of the quadratic function is likely less than the estimated average rate of change of the exponential function \( y = 6^x \), thus we denote it as \( < \).

Question 3

A) Compare \( y = 9(13)^x \) (exponential) and \( y = 9x^2 + 3x \) (quadratic) over the interval \([1,3]\):

  • Exponential function \( y = 9(13)^x \):

    • At \( x = 1 \): \( y(1) = 9(13)^1 = 117 \)
    • At \( x = 3 \): \( y(3) = 9(13)^3 = 9 \cdot 2197 = 19773 \)
    • Average rate = \( \frac{19773 - 117}{3 - 1} \) is a positive value.
  • Quadratic function \( y = 9x^2 + 3x \):

    • At \( x = 1 \): \( y(1) = 9(1)^2 + 3(1) = 12 \)
    • At \( x = 3 \): \( y(3) = 9(3)^2 + 3(3) = 90 \)
    • Average rate = \( \frac{90 - 12}{3 - 1} = \frac{78}{2} = 39 \) (positive)

Conclusion: Since both functions are increasing in that interval but change rate varies. However, the exponential function is nearly always increasing more steeply. Therefore, neither function has a negative average rate of change.

Question 4

A) Compare the estimated average rate of change for the quadratic function over the interval \([2,6]\) with the exponential function given by \( y = \frac{34(2^x)}{x} \):

We would need specific values (or an image) to calculate the slopes properly, however:

Assuming standard rates based on the behavior of quadratic vs exponential functions:

The average rate of change of the quadratic function is typically less than that of an exponential function as they grow faster.

Conclusion: Without exact calculations, I would lean towards the exponential function having a greater average rate of change in that interval based on general characteristics.

Suppose the rates were computed and found:

  • If confirmed, "The estimated average rate of change of the exponential function is greater than the estimated average rate of change of the quadratic function over the interval [2,6]."

In summary:

  1. Quadratic \( y = 4x^2 \) has a greater average rate of change.
  2. Exponential \( y = 6^x \) > Quadratic.
  3. Neither function is negative on that interval.
  4. Generally, the exponential function grows faster based on classical properties.