Chapter 2 Flashcards
Distance Formula
Suppose that P(x1, y1) and R(x2, y2) are two point in a coordinate plane. The distance between P and R, written d(P,R) is given by
√(x2-x1)^2 = (y2-y1)^2
Midpoint Formula
The midpoint M of the line segment with endpoints P(x1, y1) and Q(x2, y2) has the following coordinates. ((x1+x2)/2, (y1+y2)/2)
Center-Radius Form of the Equation of a Circle
A circle with a center (h, k) and radius r has equation (x-h)^2 + (y-k)^2 = r^2 , which is the center-radius form of the equation of the circle. A circle with center (0, 0) and radius r has equation x^2 + y^2 = r^2
General Form of the Equation of a Circle
For some real numbers c, d, and e, the equation x^2 + y^2 + cx +dy + e = 0 can have a graph that is a circle or a point, or is nonexistent.
Relation
A relation is a set of ordered pairs.
Function
A function is a relation in which, for each distinct value of the first component of the ordered pairs, there is exactly one value of the second component (x does not repeat!)
Domain and Range
In a relation consisting of ordered pairs (x, y), the set of all values of the independent variable (x) is the domain. The set of all values of the dependent variable (y) is the range.
Vertical Line Test
If every vertical line intersects the graph of a relation in no more than one point, then the relation is a function.
Increasing, Decreasing, and Constant Functions
Whenever x1 f(x2), f is decreasing.
For every x1 and x2, if f(x1) = f(x2), then f is constant.
Continuity (Informal Definition)
A function is continuous over an internal of its domain is its hand-drawn graph over that interval can be sketched without lifting the pencil from the paper.
Function Transformations
f(x) = a(x-h)^n + k
a: |a| > 1, vertical stretch; 0
Even and Odd Functions
A function f is an even function if f(-x) = f(x) for all x in the domain of f. (Its graph is symmetric with respect to the y-axis)
A function f is an odd function if f(-x) = -f(x) for all x in the domain of f. (Its graph is symmetric with respect to the origin)
Operation on Functions
Given two functions f and g, then for all values of x for which both f(x) and g(x) are defined, the functions f + g, f - g, fg, and f/g are defined as follows.
Sum: (f + g) (x) = f(x) + g(x)
Difference: (f - g) (x) = f(x) - g(x)
Product: (fg) (x) = f(x) * g(x)
Quotient: (f/g) (x) = f(x)/g(x), g(x) ≠ 0
Difference Quotient
(f(x-h) - f(x))/h
The difference quotient is a key concept in the understanding of calculus, and is related to the rate of change of a function
Composition of Functions and Domain
If f and g are functions, then the composite function, or composition, of g and f is defined by (g o f) (x) = g(f(x))
Similarly, we can state (f o g) (x) = f(g(x))
