← Precalculus syllabus

United StatesPrecalculus

Unit 4: Functions Involving Parameters, Vectors, and Matrices

14 dot points across 14 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

What are the conic sections, and how do their standard equations encode their shape and key features?

Topic 4.6 Conic Sections: identify and analyze parabolas, ellipses, circles and hyperbolas from their equations, and describe their key features.
A focused answer to AP Precalculus Topic 4.6, covering the four conic sections, their standard implicit equations, how to read center, radius, vertices and orientation from the equation, and how to tell the conics apart.
9 min answer →

What is an implicitly defined relation, and how does it differ from a function written y = f(x)?

Topic 4.5 Implicitly Defined Functions: interpret a relation given by an equation in x and y, and analyze its graph even when it is not a function of x.
A focused answer to AP Precalculus Topic 4.5, covering relations defined implicitly by an equation in x and y, why they need not pass the vertical line test, and how to analyze their graphs and extract function pieces.
9 min answer →

How does a matrix represent a linear transformation of the plane, such as a rotation, reflection or scaling?

Topic 4.12 Linear Transformations and Matrices: represent a linear transformation of the plane by a matrix, and identify the matrices for scalings, reflections and rotations.
A focused answer to AP Precalculus Topic 4.12, covering how a 2x2 matrix represents a linear transformation, how the columns are the images of the basis vectors, and the standard matrices for scalings, reflections and rotations.
9 min answer →

How is a matrix a function that takes a vector to a vector, and what do composition and inverse mean for it?

Topic 4.13 Matrices as Functions: interpret a matrix as a function from vectors to vectors, and relate matrix multiplication to composition and the inverse matrix to the inverse function.
A focused answer to AP Precalculus Topic 4.13, covering how a matrix is a function from input vectors to output vectors, how matrix multiplication corresponds to composing these functions, and how the inverse matrix undoes the transformation.
9 min answer →

How do matrices model real situations such as transitions between states over time?

Topic 4.14 Matrices Modeling Contexts: use matrices to model transitions between states, and apply repeated multiplication to project the state forward in time.
A focused answer to AP Precalculus Topic 4.14, covering how a transition matrix models movement between states, how multiplying a state vector by the matrix advances one step, and how repeated multiplication projects the system forward.
9 min answer →

What is a matrix, and how do you add, scale and multiply matrices?

Topic 4.10 Matrices: represent data with a matrix, and add, subtract, scale and multiply matrices, including multiplying a matrix by a vector.
A focused answer to AP Precalculus Topic 4.10, covering matrices as rectangular arrays, matrix addition and scalar multiplication, the row-by-column rule for matrix multiplication, and multiplying a matrix by a vector.
9 min answer →

How fast do the coordinates of a parametric curve change, and how do those rates describe the motion?

Topic 4.3 Parametric Functions and Rates of Change: compute the average rates of change of x and y with respect to t, and use them to describe the direction and relative speed of motion.
A focused answer to AP Precalculus Topic 4.3, covering the average rates of change of x and y with respect to the parameter, how their signs give the direction of motion, and how their ratio relates to the steepness of the path.
9 min answer →

How do parametric functions describe the motion of a point in the plane over time?

Topic 4.2 Parametric Functions Modeling Planar Motion: use a parametric function to model the position of a moving point over time, and describe its path, direction and position at a given time.
A focused answer to AP Precalculus Topic 4.2, covering how parametric functions model the position of a moving point over time, reading position and direction at a given time, and building a position model from a described motion.
9 min answer →

What is a parametric function, and how do x and y depend on a third variable?

Topic 4.1 Parametric Functions: define a parametric function giving x and y as functions of a parameter t, and graph and interpret the curve it traces.
A focused answer to AP Precalculus Topic 4.1, covering how a parametric function defines x and y each as a function of a parameter t, how to build a table and graph the curve, the direction of motion, and eliminating the parameter.
9 min answer →

How do you write parametric equations for a circle or a line, and how do the parameters control them?

Topic 4.4 Parametrically Defined Circles and Lines: write and interpret parametric equations for circles and lines, controlling radius, center, direction and starting point.
A focused answer to AP Precalculus Topic 4.4, covering the standard parametric forms for lines and circles, how radius, center, direction and starting point appear in the equations, and how to read or build them.
9 min answer →

How do you find a parametrization for an implicitly defined curve such as a circle or ellipse?

Topic 4.7 Parametrization of Implicitly Defined Functions: find parametric equations that trace an implicitly defined curve, and verify the parametrization satisfies the implicit equation.
A focused answer to AP Precalculus Topic 4.7, covering how to find parametric equations for an implicitly defined curve, the trig parametrization of circles and ellipses, and how to verify a parametrization satisfies the original equation.
9 min answer →

What are the determinant and inverse of a matrix, and what do they tell you?

Topic 4.11 The Inverse and Determinant of a Matrix: compute the determinant and inverse of a 2x2 matrix, and use them to determine invertibility and solve matrix equations.
A focused answer to AP Precalculus Topic 4.11, covering the determinant of a 2x2 matrix, what it measures, the inverse formula, when a matrix is invertible, and using the inverse to solve a matrix equation.
9 min answer →

What is a vector-valued function, and how does it describe position and motion in the plane?

Topic 4.9 Vector-Valued Functions: interpret a vector-valued function whose output is a position vector, and relate it to parametric motion and velocity.
A focused answer to AP Precalculus Topic 4.9, covering vector-valued functions whose output is a position vector, their equivalence to parametric functions, how to evaluate position at a time, and how average velocity is the displacement vector over time.
9 min answer →

What is a vector, and how do you add, scale and find the magnitude and direction of one?

Topic 4.8 Vectors: represent a vector by components, compute its magnitude and direction, and add, subtract and scale vectors.
A focused answer to AP Precalculus Topic 4.8, covering vectors as objects with magnitude and direction, component form, magnitude and direction angle, scalar multiplication, and vector addition and subtraction.
9 min answer →