Unit 4 · Not assessed on the exam

Functions Involving Parameters, Vectors, and Matrices

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.

What a strong answer looks like

A strong Unit 4 answer keeps track of what the parameter is doing, and distinguishes the set of points a curve occupies from the way that curve is traversed.

Practice this unit Course overview

Topics in this unit

Parametric Equations

Know

A parametrisation describes both the path a curve takes and the manner in which it is traversed. Eliminating the parameter recovers the rectangular equation but discards direction and often domain.

Apply

Solve one equation for the parameter, substitute into the other, and then state separately any restriction the parameter imposed.

Watch out

Rescaling the parameter changes the speed of traversal but not the set of points, which is easy to misread as a different curve.

Study move

Eliminate the parameter from a simple pair, then describe what the rectangular equation lost.

Implicitly Defined Curves

Know

An implicit equation constrains a relationship between the variables without expressing one as a function of the other, so it may fail the vertical line test.

Apply

Test whether an equation defines a function by asking whether any input can produce two outputs.

Watch out

A circle is a perfectly good curve but not a function of x, and the two ideas are frequently conflated.

Study move

Classify four equations by whether each defines y as a function of x, and justify each answer.

Vectors

Know

A vector carries magnitude and direction. Its magnitude is computed like a distance from its components, and addition works component by component.

Apply

Add or subtract vectors componentwise, and compute magnitude as the square root of the sum of the squared components.

Watch out

A negative component does not shorten a vector, because the component is squared before the root.

Study move

Draw a vector with a negative component, mark its two legs, and confirm the magnitude matches the calculation.

Matrix Operations

Know

Matrix multiplication is defined only when the number of columns of the first matrix equals the number of rows of the second, and the product takes the outer dimensions.

Apply

Check dimensions before computing anything, and remember that the product depends on the order.

Watch out

Matrix multiplication is not commutative, and reversing the order can make a defined product undefined.

Study move

Given two matrices of different shapes, state which orders are defined and what shape each product would have.

Determinants and Invertibility

Know

A zero determinant means a matrix is not invertible, so the system it represents has either no solution or infinitely many rather than exactly one.

Apply

Compute a two by two determinant as the difference of the diagonal products, and use it to decide whether a unique solution exists.

Watch out

A zero determinant does not by itself say which of the two failure cases applies; deciding that requires further work.

Study move

Construct a two by two system with zero determinant, then determine by inspection whether it has none or infinitely many solutions.

Linear Transformations

Know

A matrix acting on the plane moves the basis vectors, and where those two vectors land determines the entire transformation.

Apply

Identify a transformation by tracking the images of the two basis vectors and comparing against known rotations and reflections.

Watch out

A rotation and a reflection can agree on one basis vector and differ on the other, so both must be checked.

Study move

Work out where the two basis vectors go under a quarter-turn rotation, then under a reflection across the diagonal, and compare.

Emphasized in this unit

Connections and techniques that receive extra attention in this AP unit.

• Separating the set of points a curve occupies from the way the curve is traversed
• Recognising that an implicit equation need not define a function
• Checking matrix dimensions before attempting a product
• Reading a linear transformation from the images of the basis vectors

Varies by course

Related topics some schools attach to this unit and others leave out. Covered on request rather than assumed.

Coverage at all. Because this unit is not assessed, some schools teach it fully, some compress it, and some move it after the exam.
Matrix inverses. Computing an inverse by hand is included in some courses and left to linear algebra in others.
Vector applications. Physics-style force and velocity problems appear where the course is taught alongside science.

Mastery checklist

• Eliminate a parameter and say what information the rectangular equation lost.
• Decide whether an implicit equation defines y as a function of x.
• Compute a vector magnitude and add vectors componentwise.
• Determine whether a matrix product is defined and what shape it takes.
• Compute a two by two determinant and say what a zero value implies.
• Identify a rotation or reflection from the images of the basis vectors.

Check yourself

• What is lost when a parameter is eliminated?
• Why is a circle not a function of x?
• Why is matrix multiplication order-dependent?
• What does a zero determinant tell you, and what does it leave undecided?

Modeling drill

Two parametrisations trace the same circle, one with parameter t and one with 2t. Describe precisely what is the same about them and what is different, without eliminating the parameter.

ParameterImplicit equationVector magnitudeComponentDeterminantInvertibleLinear transformationBasis vector

Unit 3: Trigonometric and Polar Functions