MAT.126
P2 - P3

Slope formula: m = (y_2 - y_1)/(x_2 - x_1)

y - y_1 = m(x - x_1)

A ratio occurs when both the x and y variables have the same units and the slope therefore has no units (they cancel).A rate (or rate of change) occurs when the x and y variables have different units.Typically these rates of change are average rates of change.In either case, the result tells you the change in y per a 1 unit change in x.

y = mx + b

Ax + By + C = 0

Ax + By = D

x = a

y = b

m_1 = m_2

m_1 * m_2 = -1orm_1 = -1 / m_2

A relation between two sets X and Y is a set of ordered pairs, each of the form (x,y), where x is a member of X and y is a member of Y.The independent variable is x and the independent variable is y.The domain of the relation is X and the range of the relation is Y.A function from X to Y is a relation between X and Y that has the property that any two ordered pairs with the same x-value also have the same y-value.More formally: Let X and Y be sets of real numbers. A real-valued function f of a real variable x from X to Y is a correspondence that assigns to each number x in X exactly one number y in Y.The domain of f is the set X. The number y is the image of x under f and is denoted by f(x), which is called the value of f at x. The range of f is a subset of Y and consists of all images of numbers in X.A function is evaluated by replacing its independent variable with a specific value and then simplifying to find the value of the dependent value.An explicit equation has been solved for y. An implicit equation has not been solved for y.

These are listed with the function. They are common in application problems.

These you need to determine on your own. Remember that in order for a number to be in the domain of a function, there has to be a real-number image associated with the number.

A function from X to Y is one-to-one if to each y-value in the range there corresponds exactly one x-value in the domain.A function from X to Y is onto if its range consists of all of Y.

Identity: f(x) = xSquaring: f(x) = x^2Cubing: f(x) = x^3Square root: f(x) = sqrt(x)Absolute value: f(x) = |x|Rational: f(x) = 1/xSine: f(x) = sin xCosine: f(x) = cos x

y = f(x): Original graphy = f(x-c): Horizontal shift c units to the righty = f(x+c): Horizontal shift c units to the lefty = f(x) - c: Vertical shift c units downy = f(x) + c: Vertical shift c units upy = -f(x): Reflection (about the x-axis)y = f(-x): Reflection (about the y-axis)y = -f(-x): Reflection (about the origin)

coefficientsleading coefficientconstant termdegree

0th degree: f(x) = a (constant)1st degree: f(x) = ax + b (linear)2nd degree: f(x) = ax^2 + bx + c (quadratic)3rd degree: f(x) = ax^3 + bx^2 + cx + d (cubic)

Even degree polynomials: If a_n > 0, both ends go to positive infinity If a_n < 0, both ends go to negative infinityOdd degree polynomials: If a_n > 0, the left end goes down and the right end goes up If a_n < 0, the left end goes up and the right end goes down

Given a function f(x) for which f(a) = 0, we know that the point (a,0) is an x-intercept for the graph of f.This number a is also known as a zero (or root) of the function f.Zeros are the solutions to the equation f(x) = 0.

The function y = f(x) is even if f(-x) = f(x) (the graph is symmetric wrt the y-axis)The function y = f(x) is odd if f(-x) = -f(x) (the graph is symmetric wrt the origin)Many functions are neither even nor odd.NOTE: The only function symmetric wrt to the x-axis is f(x) = 0.