Honors Algebra 2 Final Review

Multiplying Rational Expressions

- factor numerators and combine into one -factor denominators and combine into one -simplify the expression

Multiplying Radicals with Unlike Indices

-change radicals to rational exponents -find a common denominator for the rational exponents -apply the exponent change of denominator to the radicand to create an equivalent expression -combine by multiplying coefficients and radicands -simplify the radical

Expand the logarithm

-convert power to smallest base -put positive on top and negative on bottom -roll exponents to front

simplify radicals

-factor into prime factors -group into groups of the index -for every group equal to the index move outside radical -combine outside factors together and inside together

Multiplying Radicals

-factor radicands -multiply coefficients -combine factors under one like radicand -simplify radical

Solving Radical Equations

-isolate the radical grouping -use inverse funtion to solve -check your answer

Graphing an inverse

-reflection across y=x -domain and range switch -switch x and y values

Condense a logarithm

-reorganize + and - to be together -coefficients to exponents -deal with placement order

Adding/Subtracting Radicals

-simplify each radical like terms -add of subtract the coefficients of like radicals

Dividing Imperfect Denominator Radicals

-simplify fraction under radicand -multiply by needed factors to make denominator perfect -simplify fraction as a whole

Dividing Perfect Denominator Radicals

-simplify numerator and denominator separately -simplify the fraction

Dividing Rational Expressions

-turn division problem into a multiplication problem by multiplying by the reciprocal of the dividend -factor and simplify

Exponential Decay Function

0<|b|<1

Piece-wise Function

A graph created from pieces of others -defined by their domain -domains can not overlap -if transition point of two pieces the function is continuous

Convienience Sample

A sample chosen by ease and who is close by, not typically representative.

Irrational Root Theorem

If a polynomial has rational coefficients and a+ b(sqrt c) is a zero of the equation p(x)=0 then a- b(sqrt c) is also a zero

Remainder Theorem

If p(x) is divided by (x-a) then the remainder is the number p(a) -if the remainder is 0 then (x-a) is a factor of p(x)

Rational Roots Theorem

If the polynomial p(x) has integer coefficients then every rational root of the polynomial equation p(x)=0 can be written in the form p/q

Difference of Cubes

Same Opposite Always Positive (SOAP)

Geometric Series

Sn=a1(1-r^n)/1-r

Find Inverses Algebraically

Switch x and y solve for y=x

Interquartile Range

The difference between the upper and lower quartiles.

absolute value

The distance a number is from zero on a number line

Common Ratio

The ratio of successive terms of a geometric sequence (a1/a2)

One-to-One

a function where each input only has one output and each output has only one input

Finite Graph

a function whose graph is restricted on both sides by end points

Asymptote

a line that approaches a curve at its end but does not reach -indicates defined end behavior

common logarithm

a logarithm with base 10

Synthetic Substitution

a method of evaluating polynomials by synthetic division and the remainder theorem

correlation coefficient

a number between -1 and +1 that is calculated to determine the linear dependence of two variables

Parameter

a number that describes a population

Statistic

a number that describes a sample

Inverse Relations

a relation that maps the output values of an original relation over the line y=x

Unbiased Sample

a sample representative of the entire population

Overrepresented Sample

a sample where one or more groups has a greater representation than in the general population

Underrepresented Sample

a sample where one or more groups of the population are not included

Synthetic Division

a shortcut method for dividing a polynomial by a linear factor of the form (x-a) -it can be used in place of the standard long division algorithm

Step Function

a special kind of piecewise function whose graph is a series of line segments

Census

a study of the entire population

Stratified Random

all members of designated subgroups of the population have an equal chance of being selected

Polynomial

an expression composed of terms of the sum of terms with variables as bases and positive integer exponents

Writing a Geometric Series Equation

an=a1(r)^n-1

Roots

any value of the variable that makes the equation true -aka: x-intercepts, zeros or solutions

Systematic

chosen by a rule

Removable Discontinuities

creates a hole in the graph

Non-Removable Discontinuities

creates a vertical asymptote in the graph

Composition

creating a new function using substitution using of one function into a second function

x^3

cubic

Discrete Data

data for which intermediate values do not exist, points are disconnected on a graph -domain and range written in interval notation

the radical must be greater than zero for a _________ root

even

Unbiased

every member of the population has an equal chance of being picked

Fundamental Theorem of Algebra

every polynomial of degree greater than or equal to 1 has at least one zero -where one may be a complex number

Regression

finding an equation for the graph of best fit -the shape of the graph can be linear, quadratic, exponential etc. depending on data

Neither Symmetry

graph symmetry does not lie on the y-axis or around the origin

degree of p is equivalent to degree of q

horizontal asymptote= LC/LC

degree of p less than degree of q

horizontal asymptote=0

Vertical Asymptote

if (x-a) is a factor of the denominator of a rational function but not a factor of the numerator then, x=a is the equation of a ________________ of the graph of the rational function

Complex Conjugate Theorem

if a complex number a+bi is a zero of a polynomial function then its conjugate a-bi is also a zero

Horizontal Line Test

if a function has an inverse that is also a function, then a horizontal line through the graph of the original function will pass through exactly one point

Multiplicity

if a polynomial p(x) has a multiple root at r the ____________ of r is the number of times (x-r) appears as a factor in p(x)

Even Symmetry

if graph is symmetric across the y-axis

Find a Hole

if the factor is (x-b) is in both numerator and denominator x=b

Odd Symmetry

if the graph is symmetric around the origin

zero product property

if the product of 2 or more factors is 0 the one of the factors but be 0 -if abc=0 then either a=0 or b=0 or c=0

Rational Exponents

index becomes the denominator of the exponent

x^1

linear

natural logarithm

ln(x) = loge(x)

Self Selected

members volunteer to participate

Discontinuities

occur when the denominator =0

Slant asymptote

occurs when a rational function has a denominator with a degree of 1 or higher and the degree of the numerator is exactly one more than the denominator

the radical can be any real number for a ____________ root

odd

Monomial

one term polynomial

Sample

part of a population

Cluster Sample

population divided into subgroups, then an entire subgroup is chosen at random to be the sample

x^2

quadratic

x^4

quartic

Excluded Values

real values for which a rational function is not defined

Find a Vertical Asymptote

set denominator equal to zero and solve

Extraneous Solutions

solutions that do not work with the original function -for radical functions this is because raising an even radical to a power introduces the other half of a parabola

if |bx-h|= +#

solve for two solutions

Survey

study of a sample of the population used to predict characteristics pf the entire population

if |bx-h|= -#

the answer is no solution

Mean

the average of a distribution, by adding the scores and then dividing by the number of scores

Range

the difference between the highest and lowest scores in a distribution

Population

the entire group of individuals thats is being studied

floor function

the greatest integer that is less than or equal to the given x-value

ceiling function

the least integer that is greater than or equal to the given x-value

Lower Quartile (Q1)

the median of the lower half of the data

Upper Quartile (Q3)

the median of the upper half of the data

Median

the middle score in a distribution, half the scores are above it and half are below it

Mode

the most frequently occurring score(s) in a distribution

Rational Function

the quotient of two polynomials

Degree

the sum of exponents of the variables -for polynomials it is the degree of the largest degree

Trinomial

three term polynomial

Binomial

two term polynomial

Extrapolation

using a line of best fit to estimate a value beyond the data

Interpolation

using a line of best fit to estimate a value within the data

Exponential growth function

|b| greater than 1