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Published byMeredith Herbert Modified over 2 years ago

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Proving Average Rate of Change ~adapted from Walch Education

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Key Concepts: The rate of change is a ratio describing how one quantity changes as another quantity changes. Slope can be used to describe the rate of change. The slope of a line is the ratio of the change in y-values to the change in x-values. A positive rate of change expresses an increase over time. A negative rate of change expresses a decrease over time.

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Key Concepts, continued. Linear functions have a constant rate of change, meaning values increase or decrease at the same rate over a period of time. Not all functions change at a constant rate. The rate of change of an interval, or a continuous portion of a function, can be calculated. The rate of change of an interval is the average rate of change for that period.

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Key Concepts, continued. Intervals can be noted using the format [a, b], where a represents the initial x value of the interval and b represents the final x value of the interval. Another way to state the interval is a ≤ x ≤ b. A function or interval with a rate of change of 0 indicates that the line is horizontal. Vertical lines have an undefined slope. An undefined slope is not the same as a slope of 0. This occurs when the denominator of the ratio is 0.

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Calculating Rate of Change from a Table Choose two points from the table. Assign one point to be (x 1, y 1 ) and the other point to be (x 2, y 2 ). Substitute the values into the slope formula. The result is the rate of change for the interval between the two points chosen.

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Calculating Rate of Change from an Equation of a Linear Function Transform the given linear function into slope- intercept form, f(x) = mx + b. Identify the slope of the line as m from the equation. The slope of the linear function is the rate of change for that function.

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Calculating Rate of Change of an Interval from an Equation of an Exponential Function Determine the interval to be observed. Determine (x 1, y 1 ) by identifying the starting x-value of the interval and substituting it into the function. Solve for y. Determine (x 2, y 2 ) by identifying the ending x-value of the interval and substituting it into the function. Solve for y. Substitute (x 1, y 1 ) and (x 2, y 2 ) into the slope formula to calculate the rate of change. The result is the rate of change for the interval between the two points identified.

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Remember… The rate of change between any two points of a linear function will be equal The rate of change between any two points of any other function will not be equal, but will be an average for that interval.

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Practice In 2008, about 66 million U.S. households had both landline phones and cell phones. This number decreased by an average of 5 million households per year. Use the table to the right to calculate the rate of change for the interval [2008, 2011].

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The Solution Determine (x 1, y 1 ) and (x 2, y 2 ). –(x 1, y 1 ) is (2008, 66) –(x 2, y 2 ) is (2011, 51) Using the slope formula = –5 The rate of change for the interval [2008, 2011] is 5 million households per year.

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Thanks for Watching! ~Ms. Dambreville

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