Compare the estimated average of change of the graphed quadratic function over the interval [2, 6] to the average rate of the exponential function y = 3/4 (2)^x over the same interval.
A. 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].
B. 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].
C. 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].
D. 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].
To find the estimated average rate of change of the quadratic function over the interval [2, 6], we need to find the slope of the secant line connecting the points (2, f(2)) and (6, f(6)).
The formula for the slope of a line passing through two points (x1, y1) and (x2, y2) is:
slope = (y2 - y1) / (x2 - x1)
For the quadratic function, let's assume it is in the form f(x) = ax^2 + bx + c. Plugging in x = 2 and x = 6, we get:
f(2) = 4a + 2b + c
f(6) = 36a + 6b + c
So, the slope of the secant line is:
slope_quadratic = (f(6) - f(2)) / (6 - 2)
= (36a + 6b + c) - (4a + 2b + c) / (6 - 2)
= 32a + 4b
Now, let's find the average rate of change of the exponential function y = (3/4)(2)^x over the interval [2, 6]. We need to find the slope of the secant line connecting the points (2, f(2)) and (6, f(6)).
f(2) = (3/4)(2)^2
f(6) = (3/4)(2)^6
So, the slope of the secant line is:
slope_exponential = (f(6) - f(2)) / (6 - 2)
= [(3/4)(2)^6 - (3/4)(2)^2] / (6 - 2)
= [(3/4)(64) - (3/4)(4)] / 4
= (3/4)(60) / 4
= 45/16
Comparing the two slopes, we can see that slope_quadratic = 32a + 4b and slope_exponential = 45/16.
Since we do not have information about the specific values of a and b in the quadratic function, we cannot determine which slope is greater. Therefore, the correct answer is:
C. 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].