Proving Average Rate of Change

1. Proving Average Rate of Change ~adapted from Walch Education

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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. • Solve for y. • 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.

8. 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.

9. 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].

10. 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.

11. ~Dr. Dambreville Thanks for Watching!