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# Proving Average Rate of Change - PowerPoint PPT Presentation

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.

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## PowerPoint Slideshow about ' Proving Average Rate of Change' - ivor-hutchinson

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### Proving Average Rate of Change

• 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.

• 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.

• 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.

• 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.

• 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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