Pre-Calc Final

Rectangular coordinate system

intersecting two perpendicular lines in a plane

Origin

the point of intersection

Horizontal axis

the horizontal line (the x-axis)

Vertical axis

the vertical line (the y-axis)

Quadrants

four divisions created by axes

Ordered pair

a pair of numbers (x, y) that represent a point in a coordinate plane

Midpoint

the point that is exactly halfway between two points in a coordinate plane, calculated as the average of the x-coordinates and y-coordinates.

Graph

a visual representation of data or mathematical functions plotted on a coordinate plane.

Circle

the set of all points in a plane that are equidistant from a fixed point called the center

Center

of a circle, the fixed point from which all points on the circle are equidistant.

Radius

the fixed distance from any point of the circle to the center is called the radius, which is half the diameter of the circle.

Domain

the set of x values in the ordered pairs, that a function can take as input.

Range

the set of y values in the ordered pairs, that a function can produce as output.

Function

a relationship between two quantities, where each input has exactly one output.

Scatter plot

a visual representation of a set of points

Regression line

the curve that models a set of data to show the trend in the data points.

Linear Function/Equation

an equation of the form y = f(x) = mx+b, where m and b are constants

Constant function

a function that always goes to the same value, regardless of the input. modeled by f(x) = b

Secant line

a line drawn through two points on a nonlinear curve

Average rate of change

the slope of the secant line aka the points (change in y / change in x)

Parallel lines

two distinct nonvertical lines when their slopes are equal

Perpendicular lines

two lines that their slopes are negative reciprocals (m and -1/m)

Linear functions

Constant functions

Quadratic function

Cube function

Square root function

Absolute value function

Cube root function

Reciprocal function

Symmetry

remain unchanged under reflections

Even function

if f(-x) = -f(x) for all values of x, and the graph of f is symmetric about the y-axis

Odd function

if f(-x) = -f(x) for all values of x, and the graph of f is symmetric about the origin

Piecewise-defined function

a function in which we define each “piece” on a restricted domain

Continuous

the graph of a function that has no “gaps”

Discontinuous

the graph of a function that has a gap at x=a

Increasing function

Decreasing function

Composition

the function f(g(x)) is the composition of f and g

Quadratic Functions

second degree formulas

Axis of symmetry

graph is symmetric with respect to the vertical line through the vertex

Vertex

key point on a graph or function that represents the maximum or minimum value of the function

Zeros

the values of x for which f(x) = 0, or the x-intercepts

Quadratic formula

Polynomial function

a sum of power function whose exponents are nonnegative integers

Degree

n, a non-negative integer and it cannot = 0

Coefficients

Constant term

the constant coefficient

Leading coefficient

the coefficient of the highest-powered term

Leading term

End behavior

a description of what happens as x becomes large in the negative and in the positive direction

Rational functions

a function of the form —— where p and q are polynomial functions

Horizontal asymptote

the line y=k of f if the function values get arbitrarily close to k as x gets large (either + or - or both)

Vertical asymptote

the graph of f has one at x=a if

One-to-one function

a function is if for a and b in the domain of f, a doesn’t equal b, and f(a) doesn’t equal f(b)

Horizontal line test

If there is a horizontal line that intersects a functions graph in more than one point, then the function is not one-to-one.

Inverse functions

if y = f(x) is a one-to-one function then it has one, invertible

Invertible

capable of being turned upside down, reversed in position, or subjected to inversion

Exponential function

represent change at a constant percent rate and y = f(x) has the standard form:

Exponential growth

r > 0 and b > 1

Exponential decay

r < 0 and 0 < b < 1

Natural base

base e

Compound interest

interest on interest

Logarithmic function

for b > 0 and for when b doesn’t equal 1, the log function is

Logarithms

functions for finding exponents, y

Base

b

Argument

x

Angle

formed by rotating a ray about its endpoint

Standard position

its vertex is at the origin in the xy-plane, and its initial side is the positive x-axis

Measure

the direction and the amount of rotation from the initial side to the terminal side

Degree

one unit with which to measure an angle

Radian

another type of angular measure that is better suited for applications in trig and calc

Arc length

s

Unit circle

is the circle of radius 1 centered at the origin with the equation x² + y² = 1

Trigonometric Functions

sin, cos, tan, csc, sec, cot

Identities

relationships of trig functions with each other; reciprocal identities, quotient identities, Pythagorean identities

Periodic

if there is a positive number p such that f(t+p) = f(t) for every t

Period

the smallest number p that makes f(t+p) = f(t)

Amplitude

|A|, largest value the function obtains

Phase shift

C/B, shifting cycles to the right