# Proving Average Rate of Change

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Proving Average Rate of Change

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.

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.

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.

Calculating Rate of Change from a Table
Choose two points from the table. Assign one point to be (x1, y1) and the other point to be (x2, y2). Substitute the values into the slope formula. The result is the rate of change for the interval between the two points chosen.

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.

Calculating Rate of Change of an Interval from an Equation of an Exponential Function
Determine the interval to be observed. Determine (x1, y1) by identifying the starting x-value of the interval and substituting it into the function. Solve for y. Determine (x2, y2) by identifying the ending x-value of the interval and substituting it into the function. Substitute (x1, y1) and (x2, y2) 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.

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.

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].

The Solution Determine (x1, y1) and (x2, y2).
(x1, y1) is (2008, 66) (x2, y2) is (2011, 51) Using the slope formula = –5 The rate of change for the interval [2008, 2011] is 5 million households per year.

Thanks for Watching! ~Ms. Dambreville