A Line Keeps Its Rate
Part of the Functions & Transformations section of Coddy's Algebra 2 journey. Lesson 45 of 58.
Take the line g(x) = 4x - 1. From x = 0 to x = 2, the outputs are -1 and 7, so the rate is (7 - (-1))/2. Subtracting -1 adds 1, so the top is 8 and the rate is 4. From x = 5 to x = 10, the outputs are 19 and 39, a change of 20 over an input change of 5: the rate is 4 again.
That is no accident. Every time the input grows by 1, 4x - 1 grows by exactly 4. Over any stretch of inputs the output change is 4 times the input change, and dividing by the input change leaves 4. For a line kx + c, the average rate of change is always k, its slope, whichever two inputs you pick.
Watch the double sign when an output is negative. Subtracting a negative output turns into adding, so write the bracket and work it out before you divide.
Challenge
For f(x) = 3x - 7, find the average rate of change from x = 2 to x = 6. 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