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A U-shaped upward opening curve with 2 plotted points is on a coordinate plane. The x-axis ranges from negative 4 to 7 in increments of 1. The y-axis ranges from negative 3 to 15 in increments of 1.

Compare the estimated 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)x
over the same interval.

(1 point)
Responses

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 over 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 over the interval left bracket 2 comma 6 right bracket.

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 over 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 over the interval left bracket 2 comma 6 right bracket.

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 over 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 over the interval left bracket 2 comma 6 right bracket.

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 over the interval [2,6]
.

1 answer

To compare the estimated average rates of change of the quadratic function and the exponential function over the interval \([2, 6]\), we need to perform the following calculations:

  1. Evaluate the quadratic function at \(x = 2\) and \(x = 6\).

    • Suppose the quadratic function is \(f(x) = ax^2 + bx + c\). Let's retrieve the values from the graph at these points.

  2. Calculate the average rate of change of the quadratic function: \[ \text{Average Rate of Change} = \frac{f(6) - f(2)}{6 - 2} \]

  3. Evaluate the exponential function \(y = 34(2)^x\) at \(x = 2\) and \(x = 6\):

    • Calculate \(y\) values:
      • \(y(2) = 34(2^2) = 34 \cdot 4 = 136\)
      • \(y(6) = 34(2^6) = 34 \cdot 64 = 2176\)

  4. Calculate the average rate of change of the exponential function: \[ \text{Average Rate of Change} = \frac{y(6) - y(2)}{6 - 2} = \frac{2176 - 136}{4} \]

Now we can perform the calculations:

Quadratic function:

  • Let’s assume based on the graph \(f(2)\) is around 1.5 (just an estimated value) and \(f(6)\) is around 9.5 (again, an estimated value).

Average Rate of Change of the Quadratic: \[ \text{Average Rate of Change} = \frac{9.5 - 1.5}{6 - 2} = \frac{8}{4} = 2 \]

Exponential function: \[ \text{Average Rate of Change} = \frac{2176 - 136}{4} = \frac{2040}{4} = 510 \]

Comparison:

  • The estimated average rate of change of the quadratic function is \(2\).
  • The estimated average rate of change of the exponential function is \(510\).

Now, we can conclude:

\[ 510 - 2 = 508 \]

Thus, the estimated average rate of change of the exponential function is significantly greater than that of the quadratic function.

The appropriate response would be:

  • The estimated average rate of change of the exponential function is 508 greater than the estimated average rate of change of the quadratic function over the interval [2, 6].

However, based on the options you've provided, it seems none align with this correct calculation. You may need to check the graph and reassess the values for the quadratic function, as the calculations lead to a significant divergence. If the values provided from the image or graph were different, adjust the calculation accordingly.