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=3/4(2)x over the same interval.
To find the estimated average rate of change of the graphed quadratic function over the interval [2,6], we can use the formula for the average rate of change of a function:
Average rate of change = (f(b) - f(a)) / (b - a)
Here, f(x) represents the quadratic function.
We can find the points (2, f(2)) and (6, f(6)) on the graph and plug them into the formula to calculate the average rate of change.
Let's assume that the points on the graph are (2, f(2)) = (2, 4) and (6, f(6)) = (6, 16).
Using the formula, the average rate of change of the quadratic function over the interval [2,6] is:
(16 - 4) / (6 - 2) = 12 / 4 = 3
Now, let's calculate the average rate of change of the exponential function y = (3/4)(2^x) over the same interval [2,6].
Using the formula again, we need to find the points (2, f(2)) and (6, f(6)) on the graph of the exponential function.
Calculating the points (2, f(2)) and (6, f(6)):
f(2) = (3/4)(2^2) = (3/4)(4) = 3
f(6) = (3/4)(2^6) = (3/4)(64) = 48
Using the formula, the average rate of change of the exponential function over the interval [2,6] is:
(48 - 3) / (6 - 2) = 45 / 4 = 11.25
Comparing the two average rates of change, we can see that:
Quadratic function: 3
Exponential function: 11.25
Therefore, the average rate of change of the exponential function is greater than the average rate of change of the quadratic function over the interval [2,6].