Chapter 1 and Chapter 2 Pre-Calculus

If r is the remainder obtained after a synthetic division of f(x) by (x-c), then the following statements are true

-r is the value of f(c) -If r = 0, then (x-c) is a factor of f(x) -If r = 0, then c is an x-intercept of the graph of f -If r = 0, then x = c is a solution of f(x) = 0

Constant function

A function is constant when a positive change in x results in no change in f(x)

Decreasing function

A function is decreasing when a negative change in x results in a negative change in f(x)

Increasing function

A function is increasing when a positive change in x results in a positive change is f(x)

Radical function

A function that can be written as f(x) = n√x^p

Quadratic function

A function that can be written in the form f(x) = ax2+bx+c, where a,b,and c are real numbers and a ≠ 0

What is a one-to-one function?

A function where each input corresponds to one output and each output corresponds to one input

Continuous function

A function whose graph is an unbroken line or curve with no gaps or breaks.

jump discontinuity

A graph that has discontinuity where the function moves to a different y-value and then continues. It cannot be filled in with just a point

dilation

A nonrigid transformation that has the effect of compressing or expanding the graph of a function vertically or horizontally

interval notation

A notation for describing an interval on a number line. The interval's endpoint(s) are given, and a parenthesis or bracket is used to indicate whether each endpoint is included in the interval.

set-builder notation

A notation used to describe the elements of a set

Fundamental Theorem of Algebra

A polynomial will have as many roots as its degree

Translation

A rigid transformation that has the effect of shifting the graph of a function

Horizontal translation

A shift of a graph to the left or right by h units.

Vertical translation

A shift of a graph up or down by k units.

vertical line test

A test used to determine whether a relation is a function by checking if a vertical line touches 2 or more points on the graph of a relation

Reflection

A type of rigid transformation that produces a mirror image of a graph of a function with respect to a specific line

Oblique Asymptote

An asymptote that is neither horizontal nor vertical and is sometimes called a slant asymptote. The function can pass it but approaches the value as the function apporaches infinity and negative infinity along the domain

Polynomial Inequality

An inequality that is equivalent to an inequality with a polynomial as one side and 0 as the other side

Monomial function

Any function that can be written in the form f(x) = a and f(x) = ax^n, where a and n are nonzero constant numbers

Power function

Any function that can be written in the form f(x) = ax^n, where a and n are nonzero constant numbers

Rigid transformations

Change only the position of the graph, leaving the size and shape unchanged.

Transformation

Changes that can affect the appearance of the parent graph

Composition of functions

Composition of functions f with function g is defined by f(g(x))

extrema

Critical points at which a function changes its increasing or decreasing behavior

Nonrigid transformations

Distort the shape of the graph

Factoring Polynomial Functions Over the Reals

Every polynomial function of degree n>0 with real coefficietns can be written as the product of linear factors and irreducible quadratic factors, each with real coefficients

Continuity Test

F(x) is defined at c F(X) approaches the same value from either side of x. The value that f(x) approached from each side of c is f(c).

Zero function

For the function f, any number x such that f(x) = 0

Quartic functions

Fourth degree polynomials

Conjugate Root Theorem

If a + bi is root of a polynomial with real coefficients, then a − bi is a root of that polynomial as well, where b does not equal 0

Explain the lower bound test

If c < or equal to 0 and every number in the last line of the division is alternately nonnegative and nonpositive, then c is a lower bound for the real zeroes of f

Explain the upper bound test

If c > or equal to 0 and every number of division in non-negative, then c is an upper bound for the real zeroes of f

Intermediate Value Theorem

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

Decartes' Rule of Signs

If f(x) = ax^n-1 + ax+ a is a polynomial function with real coefficients, then -the number of positive real zeros of f is equal to the number of variations in sign of f(x) or less than that number by some even number and -the number of negative real zeros of f is the same as the number of variations in sign of f(-x) or less than number by some even number

limit

If the value of f(x) approaches a unique value L as x approaches c is L

Remainder Theorem

If you divide a polynomial P(x) of degree n > 1 by x - a, then the remainder is P(a).

zeroes of a function

In other words, a "zero" of a function is an input value that produces an output of zero (0). If the function maps real numbers to real numbers, its zeroes are the x-coordinates of the points where its graph meets the x-axis. An alternative name for such a point (x,0) in this context is an x-intercept.

Transformations with absolute value

Nonrigid transformations that involve absolute value

What must exist in order for an inverse function to exist?

One relation contains (b,a) whenever the other relation contains (a,b)

Leading Coefficient Test

Positive Even- Rise left and right Negative Even- Fall left and right Positive Odd- Fall left, rise right Negative Odd- Rise left, fall right

relative vs absolute minima vs maxima

Relative minima/maxima are relative to a certain interval, whlie absolute minima/maxima are the highest or lowest value for the entirety of the function

Factor Theorem

The binomial x-r is a factor of the polynomial P(x) if and only if P(r)=0

What is true about a function's inverse if the function is one-to-one?

The domain of f is equal to the range of the inverse of f and the range of f is equal to the domain of the inverse of f

infinite discontinuity

The function at the singular point goes to infinity in different directions on the two sides.

What happens if a zero repeats an even amount of times?

The graph will be tangent to the x-axis at that point

What happens if a zero repeats an odd amount of times?

The graph will cross the x-axis at that point.

Vertical Asymptotes

The line x=c is a vertical asymptote of the graph f is the one sided limit of the point approaches either positive or negative infinity

Rational function

The quotient of two polynomial functions

Parent Function

The simplest, most general function in a family of functions.

What happens when you place absolute value bars outside any function?

The transformation reflects any portion of the graph of f(x) that is below the x-axis so that it is above the x-axis

What happens when you place absolute values outside every x value instead of the function as a whole?

The transformation results in the portion of the graph that is to the left of the y-axis being replaced by a reflection of the portion to the right of the y-axis.

Multiplicity

When a factor is repeated

roots of an equation

Where the graph would touch the x-axis

Quadratic form

Written as au^2+bu+c

removable discontinuity

a "hole" in a graph

Square root function

a function that has the form √x

piece-wise defined function

a function which is defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain

absolute value function

a function whose graph is v-shaped and contains a variable within a set of absolute value brackets

Horizontal Asymptote

a horizontal line that the curve approaches but never reaches as the curve goes to infinity and negative infinity

function

a relation in which each element of the domain is paired with exactly one element of the range

Synthetic Division

a shorthand method of dividing by a linear binomial of the form (x-a) by writing only the coefficients of the polynomials

rational inequality

any inequality that contains one or more rational expressions

leading coefficient

coefficient of the first term when the polynomial is in standard form

Extraneous Solutions

derived from an original equation that is not a solution of the original equation.

nonremovable discontinuity

describes infinite and jump discontinuities because they cannot be eliminated by redefining the function at that point

End Behavior

describes what happens to the value of f(x) as x increases or decreases without bound

odd functions

f(-x)=-f(x)

even functions

f(-x)=f(x)

reciprocal function

f(x) = 1/x x ≠ 0, the x- and y-axis are asymptotes - domain and range are set of all nonzero real numbers - no intercepts - decrease on (-∞,0) and (0,∞) - odd function

Identity Function

f(x) = x passes through all points with coordinates (a,a)

greatest integer function

f(x)=[|x|] graph= "step function"

Cubic function

f(x)=x^3 is symmetric about the origin

Discontinuous functions

functions that are not continuous

Horizontal Line Test

if a horizontal line crosses the graph of a function in more than one point, the inverse of the function is not a function

Rational Zero Theorem

if f(x) = ax^n + ... +ax + a has integer coefficients, then every rational zero of f has the following form: p/q = factor of constant term / factor of leading coefficient, both positive and negative

Linear Factorization Theorem

if f(x) is a polynomial of degree n where n>0, f has precisely n linear factors

Turning Points

indicate where the graph of a function changes from increasing to decreasing

Asymptotes

lines which the graph approaches but never reaches

average rate of change

slope of a secant line between two points

Secant line

the line through two points on a curve

Repeated zero

the related zero c of a function when a factor (x-c) occurs more than once in the completely factored form of f(x)

synthetic substitution

the use of synthetic division to evaluate a function