Summary: Rates of Change and Behaviors of Graphs

Key Equations

Average rate of change [latex]\frac{\Delta y}{\Delta x}=\frac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}[/latex]

Key Concepts

  • A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data.
  • Identifying points that mark the interval on a graph can be used to find the average rate of change.
  • Comparing pairs of input and output values in a table can also be used to find the average rate of change.
  • An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula.
  • The average rate of change can sometimes be determined as an expression.
  • A function is increasing where its rate of change is positive and decreasing where its rate of change is negative.
  • A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.
  • A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.
  • Minima and maxima are also called extrema.
  • We can find local extrema from a graph.
  • The highest and lowest points on a graph indicate the maxima and minima.

Glossary

absolute maximum
the greatest value of a function over an interval
absolute minimum
the lowest value of a function over an interval
average rate of change
the difference in the output values of a function found for two values of the input divided by the difference between the inputs
decreasing function
a function is decreasing in some open interval if [latex]f\left(b\right)a[/latex]
increasing function
a function is increasing in some open interval if [latex]f\left(b\right)>f\left(a\right)[/latex] for any two input values [latex]a[/latex] and [latex]b[/latex] in the given interval where [latex]b>a[/latex]
local extrema
collectively, all of a function’s local maxima and minima
local maximum
a value of the input where a function changes from increasing to decreasing as the input value increases.
local minimum
a value of the input where a function changes from decreasing to increasing as the input value increases.
rate of change
the change of an output quantity relative to the change of the input quantity