Algebra II Final

how would you indicate that a function does not contain a given point?

"point does not exist"

complex conjugates

(a + bi) and (a - bi) are complex conjugates; use these to rationalize complex denominators, etc.

i²

-1

i³

-i

x⁰

1

i⁴

1 (use this property to simplify!!)

how to find the inverse function

1. replace f(x) w/y 2. change the places of your x and y variables 3. solve for y; if this inverse is a function, use the notation f⁻¹(x) to notate it as such if graphing, you can sometimes find the inverse a bit easier: either reflect the graph over y=x, or swap ordered pairs to (y,x) and graph

x²/x³

1/x

x⁻²

1/x²

which way does the crocodile face for an AND inequality?

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which way does the crocodile face for an OR inequality?

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domain

All possible input values (x values)

range

All possible output values (y values)

how to write an equation, given three distinct points on the graph

N.B. always look to use intercepts if possible, since these will cancel out quickly and make the process easier!!

discontinuity

a break in the graph; use () to denote

inverse function

a function that reverses another function; f to f⁻¹

asymptote

a line that a graph approaches but never actually touches

adding/subtracting complex numbers

add/subtract the real parts; add/subtract the imaginary parts (essentially just combine the like terms)

how can you determine if something is a function?

an input cannot have multiple outputs (though an output can be the result of several different inputs); use vertical line test if in a graph form

function compositions

an operation that takes two functions and combines them so that they produce a single function; ex: f(x) and g(x) are combined such that h(x)=g(f(x))

complex number

any number that can be written in the standard form a + bi, where "a" and "b" are real numbers, and i=√(-1)

logarithm

are inverses of exponential functions

rational function

can be defined by a rational fraction (rational=fractional)

radical function

can be defined w/x (input) value in the radicand

graphing logarithmic functions

change the logarithm so that it is expressed in exponential form, then plug in points and graph from there

piecewise function

defined using different equations for different parts of the domain

square root property

equation x²=c has two solutions: x=√c and x=-√c N.B. can only be ± when solving for x!! (otherwise, it is assumed to be positive)

how to evaluate function compositions

evaluate "from the inside out" (i.e. solve the innermost function, then plug that value into the next function and evaluate, etc.)

one-to-one function

every single input has only one output; use the horizontal line test to determine this; if a function is one-to-one, then its inverse will be a function

quadratic family of functions

f(x) = ±(x-h)²+k domain: x ∈ R range: f(x) ∈ [k, ∞)

cubic family of functions

f(x) = ±(x-h)³+k domain: x ∈ R range: f(x) ∈ R

hyperbola family of functions

f(x) = ±1/(x-h)+k domain: x ∈ (-∞, h) ∪ (h, ∞) range: f(x) ∈ (-∞, k) ∪ (k, ∞)

absolute value family of functions

f(x) = ±|x-h|+k domain: x ∈ R range: [k, ∞); (-∞, k]

cube root family of functions

f(x) = ±³√(x-h)+k domain: x ∈ R range: f(x) ∈ R

square root family of functions

f(x) = ±√(x-h)+k domain: x ∈ [h, ∞) range: f(x) ∈ [k, ∞); (-∞, k]

vertex form

f(x)=a(x-h)² -the "h" values shifts the parabola left or right; a positive h= shift left, a negative h=shift right -the "k" value shifts the parabola up or down -vertex: (h,0) -axis of symmetry: x=h

how to convert standard form into vertex form

find the vertex of the graph, then plug h and k values into vertex form; express the final answer in vertex form

converting from radical notation to rational exponent notation

if m and n are positive integers, x≥0 and m/n is simplified, then: N.B. if the exponent is expressed as a decimal or a mixed number, change it to an improper fraction before evaluating normally

how to solve a quadratic inequality

might also be useful to imagine the potential graph to get a decent sense of what values will satisfy the inequality, especially w/x∈R

how to rationalize the denominator

multiply by the conjugate or by the smallest radical to effectively cancel out any radicals in the denominator

when should you use the various bracket types?

only use brackets for range/domain, everything else uses parentheses only -for DOMAIN/RANGE: parentheses (x)= excluding x; < square brackets [x]= including x; ≤ sqiggly brackets {x}= discrete notation including x; used for a single point or list of points→ {1, 4, 7}, etc. -for EVERYTHING ELSE (increasing/decreasing, constants, etc.): parentheses only!! also, these focus on x-values, not y!!

graphing exponential functions

plug in points and create a graph from there; it's a good idea to include: -y-intercept -horizontal asymptote -other significant data points -domain and range

how to write equations, given the vertex and a point on the graph

plug the values given for the vertex into vertex form for h and k; plug in the x and y values for the point given; evaluate to find the "a" value in the equation; write the final answer expressed in either vertex or standard form (double distribute for standard)

function

relationship between two variables such that each input has only one output

how to solve a quadratic inequality that contains division (aka a RATIONAL inequality)

solve by factoring and with an interval chart as you would for a quadratic inequality without division, but make sure to check for holes and test points in your interval notation to determine whether to use brackets

how to solve a linear inequality

solve normally; N.B. when dividing by a negative, make sure you flip the direction of the inequality!!

solving radical equations

some tips: -if you evaluate and get out a negative (ex: ⁴√x=-2) for a root with an EVEN index then there is no solution

how to solve a radical inequality

square both sides of the inequality to get rid of the radical (or apply another power, as necessary), and solve normally from there; make sure you also set (the term under the radical)≥0 and solve to determine the further limitations (here, splitting the equation results in an AND inequality)

index

the exponent/power that the base number is raised to; also seen as the small number on the left of the radical symbol

vertex

the minimum (if +a) or maximum (if -a)

radicand

the number under a radical symbol

factoring by completing the square

this method allows you to get both REAL and IMAGINARY solutions

factoring by the quadratic formula

this method can be used on all quadratics

multiplying/dividing complex numbers

treat "i" as a variable and distribute appropriately (see image for example)

discriminant

use b²-4ac to determine the TYPE of solution that will result

axis of symmetry

vertical line that passes through the vertex, dividing the parabola into two ≅ halves

compound interest

where: P=principal amount A=amount in account after t years R=annual rate, applied to the principal, P, n times annually

how would you write the equations for a piecewise function?

write the equations, and indicate the domain values of each of the equations with inequality statements for x; see image for ex.

(x/y)²

x²/y²

(xy)²

x²y²

x²x³

x⁵

(x²)³

x⁶

(x/y)⁻²

y²/x²

graphing complex numbers

|a+bi| → (a, b) ex: |-3+5i| → (-3, 5) use the pythagorean thm to calculate the distance of this graphed point from the origin

how to solve an absolute value equation with two absolute value terms

|x+3|=|2x+1| is equivalent to x+3=2x+1 and -x-3=2x+1 (i.e. reverse the signs of one side of the equation to find the second solution)

how to solve an absolute value inequality (when k=0)

|x|<0 is equal to x<0 and -x<0

how to solve an absolute value equation with one absolute value

|x|=k is the equivalent of x=k and x=-k

how to solve an absolute value inequality (when k>0)

|x|>k is the equivalent of an OR inequality; |x|<k is the equivalent of an AND inequality

what notation should you use to indicate an empty set?

∅

i¹

√(-1)