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.

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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…

  • 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.


~Dr. Dambreville

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