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