Unit 1
The unit that establishes the language for the rest of the course. It introduces rate of change as the thing that distinguishes function families, then works through polynomial behavior and the more delicate business of rational functions, where the graph depends on structure that is invisible until you factor.
Lesson 1.1
Average rate of change is a secant slope: output change divided by input change. It has output units per input unit and depends on the entire interval. It differs from the output difference and from an instantaneous rate.
Lesson 1.2
Finite differences compare outputs at equally spaced inputs. For a polynomial of degree n, the nth differences are constant and nonzero. A short data table can suggest a model, but other functions can interpolate the same finite set of points.
Lesson 1.3
A factor x - r produces a zero at r. Its exponent is the zero multiplicity. Odd multiplicity crosses the x-axis; even multiplicity touches and turns. Multiplicity helps connect factored equations to a graph without expanding the polynomial.
Lesson 1.4
The leading term controls polynomial end behavior because it eventually dominates all lower-degree terms. Even degree gives matching end directions; odd degree gives opposite directions. The sign of the leading coefficient determines the right-hand end.
Lesson 1.5
Factor numerator and denominator before describing a rational graph. Every original denominator zero is excluded. A canceled factor can produce a hole, while a remaining denominator zero typically produces a vertical asymptote after the expression is reduced. Cancellation does not restore an excluded input.
Lesson 1.6
Compare numerator and denominator degrees to determine rational end behavior. A smaller numerator degree gives horizontal asymptote y = 0. Equal degrees give the ratio of leading coefficients. A numerator degree exactly one larger can give a slant asymptote found by division.
Lesson 1.7
Transformations act on inputs and outputs in different ways. Subtracting h inside the function moves a graph right by h. Adding k outside moves it up by k. Multiplying outputs scales vertically and a negative output factor reflects across the x-axis.
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