Unit 6 · 7 lessons

Rates of Change & Combining Functions

How fast a function is changing — averaged over an interval or captured at a single instant — and how to build new functions by adding, multiplying, dividing and composing the ones you already know.

78 practice questions
56 flashcards
2 interactive lessons

Start lesson 1 →Unit 6 test review →

Lessons

1Average Rate of ChangeThe average rate of change is the slope of the secant line joining two points on a graph — it measures how much the output changed per unit of input across a whole interval.11 questions
2Instantaneous Rate of ChangeThe instantaneous rate of change is the slope of the tangent line at a single point — estimate it by shrinking the interval in a difference quotient until the answers stop moving.Interactive11 questions
3Rates of Change and the Shape of a GraphThe sign of the instantaneous rate of change tells you where a graph rises, where it falls and where it turns; its size tells you where it is steepest. Sinusoids change fastest at their axis, and exponentials change at a rate proportional to their value.Interactive11 questions
4Combining FunctionsAdd, subtract, multiply and divide two functions to build a new one — and work out the domain of the result from the domains of the pieces.12 questions
5Composition of Functions & ModellingFeed the output of one function straight into another to get a composite function, then use compositions to model real situations — including ones that can only be solved graphically or numerically.12 questions
6Comparing Functions and Choosing a ModelPolynomial, rational, exponential, logarithmic and sinusoidal functions side by side — domain, range, asymptotes, symmetry, end behaviour and growth — and how to choose and justify a model from a description, a table or a graph.10 questions
7Solving Equations and Inequalities Graphically and NumericallyWhen no algebra isolates $x$ , rewrite $f (x) = g (x)$ as $h (x) = f (x) - g (x) = 0$ , read the intersections with graphing technology, and pin each root down by sign changes and bisection to a stated accuracy — then read inequalities straight off the graph.11 questions