Start with how quantities change together, then dig into rates of change, zeros, end behavior, asymptotes, and holes. You'll finish by building and using polynomial and rational models.
Episode Guide — Season 1
A function is a rule that pairs each input with exactly one output. Here you learn to describe how inputs and outputs vary together: increasing, decreasing, and concave up or down.
How fast is the output changing? Average rate of change over an interval approximates the rate at a point, and its sign tells you whether the quantities rise and fall together.
Linear functions change at a constant rate, and quadratics have rates of change that are themselves linear. Compare equal-length intervals to see the pattern, and connect it to secant lines.
Where a polynomial switches between increasing and decreasing, it has a local maximum or minimum. Where its rate of change switches direction, the graph has a point of inflection.
Every polynomial of degree n has exactly n complex zeros when you count multiplicity. Use factors to find zeros, see how multiplicity shapes the graph at the x-axis, and classify functions as even or odd.
Far from the origin, the leading term takes over. Its degree and sign tell you where a polynomial's graph goes at both ends, and you'll write that using limit notation.
A rational function is a quotient of two polynomials, so its end behavior depends on which one is bigger. Compare leading terms to find horizontal and slant asymptotes.
The real zeros of a rational function come from its numerator. The zeros of both numerator and denominator mark where the function can change sign, which is how you solve rational inequalities.
When the denominator gets arbitrarily close to zero, outputs grow without bound. Learn when a zero of the denominator produces a vertical asymptote, and describe the behavior on each side.
Sometimes a factor cancels and leaves a gap instead of an asymptote. Compare multiplicities in the numerator and denominator to find holes, then locate each one with a limit.
Factored form reveals zeros, asymptotes, and holes, and standard form reveals end behavior. Rewrite expressions to get the form you need, using long division and the binomial theorem.
Add to the output or input to translate a graph, and multiply to dilate or reflect it. Combine these moves to build a new function from a parent, and track how domain and range change.
Which function type fits a situation or data set? Use rates of change, differences, and context to choose a model, then state the assumptions and domain restrictions behind it.
Build the model: use restrictions from the context, transformations of a parent function, or a regression on your calculator. Then use it to answer questions, with the right units.