AP Precalculus Unit 4 Review: Functions Involving Parameters, Vectors, and Matrices

A parametric function f(t) = (x(t), y(t)) assigns a point in the plane to each value of the parameter t. Build a table by evaluating x(t) and y(t) at several t-values, then connect the points in order of increasing t to sketch the curve. The curve has a start point at the smallest t and a terminal point at the largest t. When the function models particle motion, the graph shows the particle's trajectory, and you can identify horizontal extrema from the max and min of x(t) and vertical extrema from the max and min of y(t). Zeros of x(t) give y-intercepts of the trajectory; zeros of y(t) give x-intercepts.

As t increases, the direction of motion depends on whether x(t) and y(t) are individually increasing or decreasing: increasing x(t) means motion to the right, decreasing means left; increasing y(t) means up, decreasing means down. The average rate of change of x and y over [t1, t2] can be computed separately, and the ratio of those averages gives the slope of the curve between the two corresponding points. The same curve can be parametrized in multiple ways with different speeds or directions. A counterclockwise unit circle is (cos t, sin t) for 0 <= t <= 2pi. Transformations of this base parametrization produce any circular path: shift the center to (h, k), scale the radius to r, and adjust the angular frequency. A line segment from (x1, y1) to (x2, y2) can be parametrized using an initial position and constant rates of change for x and y.

An equation in two variables like x^2 + y^2 = 25 implicitly defines a curve. You can graph it by finding solution pairs, and you can solve for one variable to extract explicit function branches. If nearby points on the curve have a positive ratio of changes, both variables increase or decrease together; a negative ratio means one increases as the other decreases. Conic sections are the main implicit curves in this unit. Parabolas use vertex form y - k = a(x - h)^2 or x - h = a(y - k)^2. Ellipses use (x - h)^2/a^2 + (y - k)^2/b^2 = 1, with circles as the special case a = b. Hyperbolas use (x - h)^2/a^2 - (y - k)^2/b^2 = 1 (opening left/right) or (y - k)^2/b^2 - (x - h)^2/a^2 = 1 (opening up/down), with asymptotes y - k = +/-(b/a)(x - h). Any implicit curve can be converted to parametric form: if you can solve for y, use (t, f(t)); if you can solve for x, use (f(t), t). Ellipses use x(t) = h + a cos t, y(t) = k + b sin t. Hyperbolas opening left/right use x(t) = h + a sec t, y(t) = k + b tan t.

A vector is a directed line segment described by its components (a, b), where a = x2 - x1 and b = y2 - y1. Its magnitude is sqrt(a^2 + b^2). Scalar multiplication k*(a, b) = (ka, kb) produces a parallel vector. Vector addition (a1 + a2, b1 + b2) can be visualized tip-to-tail. The dot product of (a1, b1) and (a2, b2) is a1a2 + b1b2, which equals the product of the magnitudes times the cosine of the angle between them. If the dot product is zero, the vectors are perpendicular. A unit vector in the direction of (a, b) is (a, b) divided by its magnitude. The standard unit vectors are i = (1, 0) and j = (0, 1). A vector-valued function p(t) = (x(t), y(t)) gives the position of a particle at time t. The velocity vector is v(t) = (x'(t), y'(t)), and the speed is its magnitude. The sign of x'(t) tells you left or right; the sign of y'(t) tells you up or down.

An n x m matrix has n rows and m columns. Two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second; the result is an n x p matrix if the first is n x m and the second is m x p. Each entry in the product is the dot product of a row from the first matrix and a column from the second. The identity matrix I has 1s on the main diagonal and 0s elsewhere; multiplying any square matrix by I returns the original. The inverse A^(-1) of a 2x2 matrix A = [[a, b],[c, d]] exists when det(A) = ad - bc is not zero, and equals (1/(ad-bc)) * [[d, -b],[-c, a]]. If det(A) = 0, the matrix is singular and has no inverse. The absolute value of the determinant also gives the area of the parallelogram spanned by the two column (or row) vectors of the matrix.

A linear transformation L from R^2 to R^2 maps every input vector to an output vector such that each output component is a sum of constant multiples of the input components. Every such transformation corresponds to a unique 2x2 matrix A where L(v) = Av. To find the matrix, check where the unit vectors i = (1, 0) and j = (0, 1) land; those images become the columns of A. The rotation matrix [[cos t, -sin t],[sin t, cos t]] rotates every vector counterclockwise by angle t. The absolute value of the determinant of A gives the factor by which the transformation scales areas. Composing two linear transformations corresponds to multiplying their matrices: if L1 has matrix A and L2 has matrix B, then L2 composed with L1 has matrix BA. The inverse transformation L^(-1) is given by A^(-1) when det(A) is not zero.

A transition matrix encodes the rates at which a system moves between two states over discrete time steps. Each entry represents the fraction of a state that transitions to another state. Multiplying the transition matrix by a state vector gives the predicted state after one step. Repeating this multiplication predicts future states. As the number of steps grows, the state vector approaches a steady state, a fixed distribution that does not change under further multiplication. The inverse of the transition matrix can be used to work backward and recover past states from a known current state.