Finding Average Rate of Change:
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Identify the Interval: You need to know the starting and ending points (often in terms of x-values) for which you want to calculate the average rate of change. Let's call these points A (x₀, y₀) and B (x₁, y₁).
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Use the Formula: The average rate of change (A) is calculated using the following formula:
A = (y₁ - y₀) / (x₁ - x₀)
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A represents the average rate of change.
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y₁ and y₀ are the function's output values (y-coordinates) at points B and A respectively.
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x₁ and x₀ are the input values (x-coordinates) at points B and A respectively.
Essentially, you're subtracting the starting value (y₀) from the ending value (y₁) and dividing that by the change in the input value (x₁ - x₀).
Example:
Imagine you have a function representing temperature over time. You want to find the average rate of change in temperature between 2pm (x₀ = 2) and 4pm (x₁ = 4). Let's say the temperature at 2pm was 20°C (y₀ = 20) and at 4pm it was 25°C (y₁ = 25).
A = (25 - 20) / (4 - 2) = 5 / 2 = 2.5°C/hour
The average rate of change in this case is 2.5°C per hour. So, on average, the temperature increased by 2.5°C between 2pm and 4pm.
Visualizing with a Graph:
Imagine the graph of the function. The average rate of change represents the slope of a secant line drawn between points A and B on the graph. This line cuts through the curve of the function, and its slope tells you the average rate of change over that interval.
Remember:
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This method calculates the average rate of change over an interval, not the instantaneous rate of change at a specific point (which requires calculus).
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The average rate of change can be positive (increasing), negative (decreasing), or zero (constant).