Average Rate of Change Quadratics

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What is the average rate of change of a function?

Back

The average rate of change of a function over an interval [a, b] is given by the formula: \( \frac{f(b) - f(a)}{b - a} \). It represents the change in the function's value divided by the change in the input value.

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How do you calculate the average rate of change for the function f(x) = 2x^2 + 12x + 16 on the interval [-3, -2]?

Back

1. Calculate f(-3) = 2(-3)^2 + 12(-3) + 16 = 2(9) - 36 + 16 = -2. 2. Calculate f(-2) = 2(-2)^2 + 12(-2) + 16 = 2(4) - 24 + 16 = -4. 3. Average rate of change = \( \frac{f(-2) - f(-3)}{-2 - (-3)} = \frac{-4 - (-2)}{1} = -2 \).

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What is the average rate of change of a function on an interval where the function is undefined?

Back

If the function is undefined on the interval, the average rate of change cannot be calculated as it requires valid function values at both endpoints.

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What is the average rate of change of the function f(x) = x^2 from x = 0 to x = 1?

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1. Calculate f(0) = 0^2 = 0. 2. Calculate f(1) = 1^2 = 1. 3. Average rate of change = \( \frac{f(1) - f(0)}{1 - 0} = \frac{1 - 0}{1} = 1 \).

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What does a negative average rate of change indicate about a function on an interval?

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A negative average rate of change indicates that the function is decreasing over that interval.

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What is the average rate of change of the function f(x) = -x^2 + 4 on the interval [1, 3]?

Back

1. Calculate f(1) = -1^2 + 4 = 3. 2. Calculate f(3) = -3^2 + 4 = -5. 3. Average rate of change = \( \frac{f(3) - f(1)}{3 - 1} = \frac{-5 - 3}{2} = -4 \).

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What is the significance of the average rate of change in real-world applications?

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The average rate of change can represent speed, growth rate, or any change in quantity over time, making it useful in fields like physics, economics, and biology.

Front

How do you find the average rate of change from a graph?

Back

To find the average rate of change from a graph, identify two points on the curve, (x1, y1) and (x2, y2), and use the formula: \( \frac{y2 - y1}{x2 - x1} \).

Front

What is the average rate of change of the function f(x) = 3x - 2 from x = 2 to x = 5?

Back

1. Calculate f(2) = 3(2) - 2 = 4. 2. Calculate f(5) = 3(5) - 2 = 13. 3. Average rate of change = \( \frac{f(5) - f(2)}{5 - 2} = \frac{13 - 4}{3} = 3 \).

Front

What is the average rate of change of a constant function?

Back

The average rate of change of a constant function is always 0, as the function's value does not change regardless of the input.