Use the image to answer the question.

Question

A curve with 2 plotted points is on a coordinate plane. The x-axis ranges from negative 10 to 4 in increments of 1. The y-axis ranges from negative 4 to 18 in increments of 1. the points are: (-9,-1) and (2,15)

Compare the magnitude of the estimated average rates of change of the exponential function pictured above and the quadratic function f(x)=x^2−20 over the interval [−9,2] and identify which function has a greater rate of change than the other.

(1 point)
Responses

The estimated average rate of change of the exponential function over the interval [−9,2]
is negative, while the estimated average rate of change of f(x)
over the interval [−9,2]
is positive.
The estimated average rate of change of the exponential function over the interval left bracket negative 9 comma 2 right bracket is negative, while the estimated average rate of change of f of x over the interval left bracket negative 9 comma 2 right bracket is positive.

The estimated average rate of change of both the exponential function and f(x)
over the interval [−9,2]
is approximately 1.5
.
The estimated average rate of change of both the exponential function and f of x over the interval left bracket negative 9 comma 2 right bracket is approximately 1 point 5.

The magnitude of the estimated average rate of change of the exponential function over the interval [−9,2]
is greater than the magnitude of the estimated average rate of change of f(x)
over the interval [−9,2]
.
The magnitude of the estimated average rate of change of the exponential function over the interval left bracket negative 9 comma 2 right bracket is greater than the magnitude of the estimated average rate of change of f of x over the interval left bracket negative 9 comma 2 right bracket.

The magnitude of the estimated average rate of change of the exponential function over the interval [−9,2]
is less than the magnitude of the estimated average rate of change of f(x)
over the interval [−9,2]
.

Answer 1

To compare the average rates of change of the exponential function and the quadratic function \( f(x) = x^2 - 20 \) over the interval \([-9, 2]\), we need to calculate the average rate of change for each function over that interval.

1. Quadratic Function \( f(x) = x^2 - 20 \):

   • Calculate \( f(-9) \) and \( f(2) \): \[ f(-9) = (-9)^2 - 20 = 81 - 20 = 61 \] \[ f(2) = (2)^2 - 20 = 4 - 20 = -16 \]
   • The average rate of change from \(-9\) to \(2\) for \( f(x) \): \[ \text{Average rate of change} = \frac{f(2) - f(-9)}{2 - (-9)} = \frac{-16 - 61}{2 + 9} = \frac{-77}{11} = -7 \]

2. Exponential Function:

   • Let's assume the points given \((-9, -1)\) and \((2, 15)\) correspond to the exponential function \(g(x)\).
   • The average rate of change from \(-9\) to \(2\) for the exponential function: \[ \text{Average rate of change} = \frac{g(2) - g(-9)}{2 - (-9)} = \frac{15 - (-1)}{2 + 9} = \frac{16}{11} \approx 1.45 \]

Now, with both average rates of change calculated:

• The estimated average rate of change of \( g(x) \) (exponential) over the interval \([-9, 2]\) is approximately \( 1.45 \).
• The estimated average rate of change of \( f(x) \) (quadratic) over the interval \([-9, 2]\) is \( -7 \).

Comparison:

• The estimated average rate of change of the exponential function is positive, while the estimated average rate of change of the quadratic function is negative.

Therefore, the correct response is: "The estimated average rate of change of the exponential function over the interval \([-9,2]\) is negative, while the estimated average rate of change of \( f(x) \) over the interval \([-9,2]\) is positive."