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Unit 4
Not on the AP examThis unit adds new ways to describe motion and change: parametric functions, implicitly defined curves like circles and ellipses, vectors, and matrices. It's taught in some AP Precalculus classes but isn't tested on the AP exam. It's great preparation for calculus, physics and computer graphics.
Longer videos that cover the whole unit. Good for a first pass or a final review.
A parametric function gives both x and y in terms of a third variable, t, called the parameter. You make a table of (x(t), y(t)) values and plot them in order of increasing t to trace the curve.
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Parametric functions can describe a particle moving in a plane, with (x(t), y(t)) its position at time t. The farthest left, right, up and down it goes come from the maximum and minimum values of x(t) and y(t).
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If x(t) is increasing the particle moves right, and if y(t) is increasing it moves up. Dividing y's average rate of change by x's gives the slope of the secant line between two points on the curve.
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x = cos t, y = sin t traces the unit circle counterclockwise, and transformations of it give any circle. A line segment can be traced by starting at one point and adding constant rates of change for x and y.
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An equation in x and y, like x² + y² = 25, can describe a curve without giving y as a single function of x. Solving for one variable can give one or more functions that make up parts of the curve.
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Parabolas, ellipses and hyperbolas are curves you get by slicing a cone, and each has a standard equation in x and y. From the equation you'll read the vertex or center, the lengths of the axes, and which way the curve opens.
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You can turn an implicit curve into a parametric one. For example, x = h + a cos t, y = k + b sin t traces an ellipse, and sec t and tan t can trace a hyperbola.
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A vector has magnitude (length) and direction, written in components as ⟨a, b⟩. You'll add vectors, multiply them by a scalar, find unit vectors, and use the dot product to find the angle between two vectors.
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A vector-valued function p(t) = ⟨x(t), y(t)⟩ gives a position vector for each time t. A similar function can describe velocity, showing which direction and how fast a particle is moving.
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A matrix is a rectangular grid of numbers with rows and columns. You can multiply two matrices when the first has as many columns as the second has rows; each entry of the product is a row times a column.
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For a 2×2 matrix [a b; c d], the determinant is ad − bc, and its absolute value gives the area of the parallelogram formed by the row or column vectors. A square matrix has an inverse exactly when its determinant isn't zero.
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A linear transformation maps vectors to vectors, and every one in the plane can be written as multiplying by a 2×2 matrix. Because of that, it always leaves the origin ⟨0, 0⟩ where it is.
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Where a matrix sends the unit vectors ⟨1, 0⟩ and ⟨0, 1⟩ tells you its columns. Matrices can rotate or dilate the plane; composing two transformations means multiplying their matrices, and the inverse matrix undoes the transformation.
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A transition matrix can model how a population moves between states over time, like customers switching brands. Multiplying by the matrix again and again predicts future states, and the inverse can estimate earlier ones.
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