To find the average rate of change of both the quadratic function and the exponential function over the interval [2,6], we can use the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \( f(b) \) and \( f(a) \) are the function values at the endpoints of the interval.
Let's denote:
- \( f(x) \) as the quadratic function,
- \( g(x) = \frac{3}{4} \cdot 2^x \) as the exponential function.
Analyzing the Quadratic Function \( f(x) \):
Since we don't have the specific values for the quadratic function from the image, let's denote the estimated values at \( x = 2 \) and \( x = 6 \):
- Let \( f(2) \) be the value at \( x = 2 \),
- Let \( f(6) \) be the value at \( x = 6 \).
The average rate of change for the quadratic function is: \[ \text{Average Rate of Change for } f(x) = \frac{f(6) - f(2)}{6 - 2} \]
Analyzing the Exponential Function \( g(x) \):
Calculating \( g(2) \) and \( g(6) \):
- For \( g(2) = \frac{3}{4} \cdot 2^2 = \frac{3}{4} \cdot 4 = 3 \)
- For \( g(6) = \frac{3}{4} \cdot 2^6 = \frac{3}{4} \cdot 64 = 48 \)
So the average rate of change for the exponential function is: \[ \text{Average Rate of Change for } g(x) = \frac{48 - 3}{6 - 2} = \frac{45}{4} = 11.25 \]
Conclusion:
The last task is to compare the two average rates of change.
Without the specific values for the quadratic function, we can't calculate its average rate of change. However, if we let’s say the average rate of change of the quadratic function over this interval was found to be \( x \) (where \( x \) is calculated based on graph or information you have), we would compare this to the exponential function’s average rate of change:
Now you would accurately choose one of the options based on that comparison.
Since I cannot visualize the plotted points from the image, you will need to analyze it to find \( f(2) \) and \( f(6) \), compute the average rate of change for \( f(x) \), and finally conclude which of the statements best describes the relationship between the two rates of change over the interval [2, 6].