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Proving Average Rate of ChangePowerPoint Presentation

Proving Average Rate of Change

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### Proving Average Rate of Change

~adapted from Walch Education

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.
- 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.

Remember… of an Exponential

- 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 of an Exponential

- 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 of an Exponential

- 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.

~Dr. of an Exponential Dambreville

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