To compare the estimated average rate of change of the quadratic function and the exponential function over the interval [2,6], we need to calculate the average rate of change for both functions.
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Average Rate of Change Formula:
The average rate of change of a function \(f\) over an interval \([a, b]\) is given by:
\[ \frac{f(b) - f(a)}{b - a} \] -
Quadratic Function:
Let's denote the quadratic function as \(f(x)\). You need to obtain the values at \(x=2\) and \(x=6\) from the graph:- \(f(2) = y_1\) (value from the graph at \(x=2\))
- \(f(6) = y_2\) (value from the graph at \(x=6\))
Then, compute the average rate of change:
\[ \text{Average Rate of Change (quadratic)} = \frac{y_2 - y_1}{6 - 2} = \frac{y_2 - y_1}{4} \]
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Exponential Function:
The exponential function is given as \(y = 34(2)^x\). Plug in the values for \(x=2\) and \(x=6\):- \(y(2) = 34(2)^2\)
- \(y(6) = 34(2)^6\)
Calculate these values:
\[ y(2) = 34 \cdot 4 = 136 \]
\[ y(6) = 34 \cdot 64 = 2176 \] Now calculate the average rate of change for the exponential function: \[ \text{Average Rate of Change (exponential)} = \frac{2176 - 136}{6 - 2} = \frac{2040}{4} = 510 \]
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Comparison:
Now compare the average rate of change of both functions. If you have the quadratic function’s average rate of change (let's call it \(AR_{quad}\)), you can determine:- If \(AR_{quad}\) is significantly lower than \(510\), the difference will indicate which one is greater.
Given the information and options, it's likely you will find that the exponential function has a much higher average rate of change than the quadratic function over that interval, leading to a response along the lines of: "The estimated average rate of change of the exponential function is significantly greater than the estimated average rate of change of the quadratic function over the interval [2,6]."
Given the choices, you would need the specific numerical results from the graph to fill in the exact values in the response.