APPC pgs 1-5

slope

Represents the number of unit the line rises or falls for each unit of horizontal change from left to right

point-slope form

𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

slope-intercept form

y=mx+b

general form

Ax+By+C=0

parallel lines

Two distinct nonvertical lines that have the same slope

perpendicular lines

Two nonvertical lines whose slopes are negative reciprocals of each other

function

Relation that assigns each element in set A to exactly one element in set B

domain

The elements of set A (sometimes called x-values or inputs)

range

The elements of set B (sometimes called y-values or outputs)

independent variable

The value that you can choose (sometimes called the domain, x-values, inputs)

dependent variable

The value that you cannot choose (sometimes called the range, y-values, outputs)

function notation

An equation of y in terms of x can be rewritten so that y=f(x).

piecewise-defined function

A function that is defined using two or more expressions for different intervals of the domain

implied domain

In a function with an unspecified domain, the set of all real numbers for which the expression used to define the function is real

Vertical Line Test

A visual test used to determine if a graph represents a function. To be a function the vertical line can only touch graph at most one time

Increasing Function

Describes a function f in which for any two points, a positive change in x results in a positive change in f(x). As x moves to the right the y gets bigger

Decreasing Function

Describes a function f in which for any two points, a positive change in x results in a negative change in f(x). As x moves to the right the y gets smaller

Constant Function

Describes a function f in which for any two points, a positive change in x results in a zero change in f(x)

Maximum

For a function f, the greatest value of f(x). A critical point on the graph of a function where the curve changes from increasing to decreasing

Minimum

For a function f, the least value of f(x). A critical point on the graph of a function where the curve changes from decreasing to increasing

Even Function

A function that is symmetric with respect to the y-axis

Odd Function

A function that is symmetric with respect to the origin

Linear Function

A function of the form 𝑓(𝑥) = 𝐶

Quadratic Function

A function of the form f(x)=ax2 +bx+c, where a is not 0, with parent function f(x)=x^2

Cubic Function

A function of the form f(x)=ax^3 +bx^2 +cx+d, where a is not 0, with parent function f(x)=x^3

Square Root Function

A function that contains a square root of the independent variable, with parent

function f (x) = *square root* x

Absolute Value Function

A function that contains an absolute value of the independent variable, with the parent function of f (x) = |x|

Composite Function

The combining of functions by using the result of one function to evaluate a second function. The composition of function f with function g is defined by [f o g](x)= f[g(x)]

Inverse Function

Two functions f and f^-1 are inverse functions if and only if f[f^-1(x)]=x for every x in the domain of f^-1(x) and f^-1[f(x)]=x for every x in the domain of f(x)

One-To-One

A function in which no x-value is matched with more than one y-value and no y-value is matched with more than one x-value

axis of symmetry

A line about which a figure is symmetric (mirror image about a line). Found by x = -b/2a

standard form of a quadratic

A function of the form 𝑓(𝑥) = 𝑎(𝑥 − ℎ)²+k

parabola

a “U” shaped graph

continuous function

A function that can be graphed with no breaks, holes, gaps

power function

A function of the form f (x) =ax^n where a and n are nonzero real numbers

leading coefficient test

a. even exponent of leading coefficient

i. leading coefficient > 0 up/up

ii. leading coefficient < 0 down/down

b. odd exponent of leading coefficient

iii.leading coefficient > 0 down left/up right

iv.leading coefficient < 0 up left/down right

extrema

The maximum and minimum values of a function

zeros of a function

the x-intercepts of the graph of a function (where the graph crosses the x-axis)

Intermediate Value Theorem

If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then there exists at least one number c in [a,b] such that f(c) =k

improper rational expression

Degree of numerator greater than degree of denominator

Proper rational expression

Degree of numerator less than degree of denominator

Synthetic Division

A shortcut for dividing a polynomial by a linear factor of x-c

Remainder Theorem

If a polynomial f(x) is divided by x-c, the remainder is r=f(c)

Factor Theorem

A polynomial f(x) has a factor (x-c) if and only if f(c)=0

Rational Zero Theorem

Describes how the leading coefficient and constant term of a polynomial function with integer coefficients can be used to determine a list of all possible rational zeros

Descartes’ Rule of Signs

A rule that gives information about the number of positive and negative real zeros of a polynomial function by looking at a polynomial’s variations in sign

Additive Identity (Complex Number System)

The number 0 (you can add this to any number and NOT change its identity)

Additive Inverse (Complex Number System)

A number in the form -a-bi, where b is not equal to 0

Standard Form (Complex Number System)

A complex number written in the form a+bi

Real Part (Complex Number System)

In an imaginary number a+bi, a is the real part

Imaginary Part (Complex Number System)

In an imaginary number a+bi, b is the imaginary part

Imaginary Number (Complex Number System)

Another name for a complex number of the form a+bi, where b is not equal to 0

Pure Imaginary Number (Complex Number System)

An imaginary number (a+bi) where a=0 (which means no real part exists)

Complex Conjugates (Complex Number System)

Two complex numbers of the form a+bi and a-bi

Fundamental Theorem of Algebra

If f(x) is a polynomial of degree n, where n > 0, then f has at least one zero in the complex number system

Vertical Asymptote

The line x=c is a vertical asymptote of the graph f if lim x→ c- f(x)=±infinity or

lim x→ c+ f (x) = ±infinity

Horizontal Asymptote

The line y=c is a vertical asymptote of the graph f if lim x→ c- f(x)= c or

lim x→ c+ f (x) = c