A Curve Changes Rate
Part of the Functions & Transformations section of Coddy's Algebra 2 journey. Lesson 46 of 58.
A curve does not keep one rate. Take g(x) = x^2. From x = 0 to x = 2 the output goes from 0 to 4, so the rate is 4 over 2, which is 2. From x = 2 to x = 4 the output goes from 4 to 16, so the rate is 12 over 2, which is 6. Same width of interval, three times the rate.
The graph shows why. To the right of 0 the parabola gets steeper as x grows, so the line through two points further right is steeper too. The rate belongs to the interval, not to the function, so always say which interval you mean.
An interval can even give a rate of 0. From x = -2 to x = 2 the outputs are 4 and 4: the curve went down and came back up to the same height, so on average it did not change at all.
Challenge
For f(x) = x^2, find the average rate of change from x = 2 to x = 5. Use the rule on both calls, then type the arithmetic line by line.
Try it yourself
Simplify the expression
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Functions & Transformations
4Combining Functions
Adding FunctionsSubtracting FunctionsMultiplying FunctionsAt a Single NumberRecap: Combining10Functions in Stories
Writing the RuleUsing the RuleWorking BackwardUndoing a ConversionRecap: Stories2Expressions as Inputs
A Letter as InputA Sum as InputShifting the InputSolving f(x) = kRecap: Inputs as Expressions5Composition
One Inside AnotherOrder MattersA Function of ItselfComposing RulesBreaking ApartRecap: Composition8Stretching and Flipping
Taller and FlatterFlipping OverFlip, Then ShiftReading the RuleRecap: Stretch and Flip6Inverse Functions
Undoing a FunctionSwap and SolveFractions in InversesChecking an InverseA Value of the InverseRecap: Inverses9Rate of Change
Average Rate of ChangeA Line Keeps Its RateA Curve Changes RateVelocity as a RateRecap: Rate of Change