Unit 4
Taught as part of the course but never tested. It covers parametric and implicitly defined curves, vectors, and matrices as transformations. Worth learning for later courses in calculus and linear algebra, but it should not consume revision time in the weeks before the exam.
Additional course content; not assessed on the AP Precalculus Exam.
Lesson 4.1
Parametric equations express both coordinates in terms of a shared parameter. Eliminating the parameter gives a relation, but the parameter interval and direction must be retained. Different parameterizations can trace the same curve at different speeds.
Lesson 4.2
An implicit equation relates coordinates without necessarily giving one output for each input. Test whether a vertical line can meet the curve more than once. Choosing a branch can create a function but also restricts its range.
Lesson 4.3
A vector records magnitude and direction. Add components to combine displacements; multiply all components to scale. A position vector and a point may use the same coordinate pair, but they represent different objects.
Lesson 4.4
Add matrices entry by entry when dimensions agree. Matrix multiplication uses row-column dot products and requires the inner dimensions to match. Multiplication order generally matters.
Lesson 4.5
A two-by-two determinant measures signed area scaling and detects invertibility. A zero determinant means a linear transformation collapses the plane to a line or point. Divide the adjugate by the determinant only when it is nonzero.
Lesson 4.6
A matrix maps column vectors to column vectors. Its columns are the images of the standard basis vectors. Linear transformations preserve the origin; a translation by a nonzero vector is not linear.
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