Matrix multiplication combines each row of the first matrix with each column of the second. The entry in row i, column j of AB is the sum of the products of row i of A with column j of B. You can multiply an m × n matrix by an n × p matrix, and the product is m × p. Order matters: usually AB ≠ BA.
A matrix is a rectangular array of numbers arranged in rows and columns. Basic matrix operations are adding, subtracting and multiplying by a number (a scalar). To add or subtract, the matrices must have the same dimensions, and you combine the entries in matching positions. To multiply by a scalar, multiply every entry by that number.
A logistic function models growth that starts out nearly exponential, then slows down and levels off at a maximum called the carrying capacity. Its common form is P(t) = C/(1 + ae^(−kt)), where C is the carrying capacity, k > 0 sets the growth speed, and P(0) = C/(1 + a). Growth is fastest when P = C/2.
Exponential regression finds the exponential function y = a · b^x that best fits a set of data points. A graphing calculator does it with the ExpReg command, giving the starting value a and the growth factor b. Use it when the data grows or decays by a roughly constant factor, then use the model to describe the trend and make predictions.
A semi-log plot is a graph with a logarithmic scale on the y-axis and a regular scale on the x-axis. Exponential data, y = a · b^x, makes a straight line on it, because log y = log a + x log b is linear in x. The line’s slope is log b and its intercept is log a, so you can read the model from the line.
A logarithmic scale spaces numbers by powers of 10, so each equal step on the scale multiplies the quantity by 10. pH, decibels and earthquake magnitude all use log scales. A difference of d units on a base-10 log scale means one quantity is 10^d times the other, so pH 3 is 10^3 = 1000 times more acidic than pH 6.
Partial fraction decomposition splits a rational expression into a sum of simpler fractions, the reverse of adding fractions with a common denominator. For example, (5x − 1)/(x^2 − 1) = 2/(x − 1) + 3/(x + 1). Factor the denominator, write one fraction for each factor with unknown numerators, then solve for those numerators.
To solve rational inequalities like (x − 3)/(x + 2) > 0, get 0 on one side and write the other side as a single fraction. Find the critical values, where the numerator is 0 or the denominator is 0. Test each interval between them on a sign chart, and never include a value that makes the denominator 0.
A slant asymptote (or oblique asymptote) is a slanted line y = mx + b that the graph of a rational function approaches as x goes to infinity or negative infinity. It happens when the numerator’s degree is exactly one more than the denominator’s. To find it, divide the numerator by the denominator; the quotient, without the remainder, is the asymptote.
To solve polynomial inequalities like (x − 1)(x + 2)(x − 4) > 0, move every term to one side, factor, and find the zeros. The zeros split the number line into intervals where the polynomial keeps one sign. Test one number in each interval with a sign chart, then keep the intervals that make the inequality true.