Master Algebra 2 (except 11.4 and half of chp. 7)
Natural Numbers
"Counting numbers" (1,2,3,...)
The solutions of quadratic equations are
"Zeros" or x-intercepts
What does the function operation of addition look like?
( f + g) (x) = f(x) + g(x)
What does the function operation of subtraction look like?
( f - g)(x) = f(x) - g(x)
What does the function operation of division look like?
( f/g)(x) = f(x)/g(x) where g(x) cannot equal 0
What is the y-intercept of a quadratic in standard form?
(0,c) -plug 0 in for x
What is the most important equation in completing squares and when do you use it?
(B/2)" -- used by adding to both sides in order to make equation into perfect square trinomial (after equation is rewritten)
What does the function operation of multiplication look like?
(fxg)(x) = f(x) x g(x)
You can never simplify radicals if
+ or - sign in middle and no exponents
If even root and it simplifies, remember
+-
What are the degree specifications when it comes to horizontal and slant asymptotes?
- If deg of num is less than deg of denom, HA at y=0 - If deg of num is equal to deg of denom, divide leading coefficients (n/d), HA at y=quotient - If deg of num is greater than deg of denom, divide restriction into coeffs of num, slant asym
If you flip what is being subtracted in a phrase, what must you leave behind?
-1
What is i"?
-1
When putting years into a calculator graph, start with
0 rather than the year value
The polynomial you get as a result of the (n+1) Point Principle should have an r value that equals
1. If it doesnt, try Polynomials of lesser degrees
How do you complete the square w/ a coefficient?
1. Ax" + bx = c (vars on one side, # on other) 2. Factor out leading coefficient 3. Add (b/2)" onto left side (NOTE: b is after factored version!) 4. Multiply (b/2)" by factored leading coefficient - add THAT to RIGHT side 5. Factor now perfect square trinomial 6. Divide both sides of equation by factored coefficient (left side--it goes away. Right side--it's divided by) 7. Take square roots of both sides - To take square roots of fractions, root top and bottom separately -to get radical out of denim, multiply whole fraction by bottom radical over itself (rationalize it) - denim becomes itself w/o radical, numerator gets multiplied by denominator 8. Solve for x
How do you solve a perfect square trinomial?
1. Factor into a binomial squared 2. Take square roots of both sides -REMEMBER plus or minus! Could also be solved just by simple factoring
What are the six factoring techniques?
1. Factor out a GCF 2. Quadratic Trinomial 3. Perfect Square Trinomial 4. Différence of perfect squares 5. Use grouping 6. Sum or difference of cubes
How do you solve an equation by taking a square root?
1. First get x" by itself 2. Take square roots of both sides - x" will become x - reduce other side AND REMEMBER PLUS OR MINUS
How do you solve basic radical equations?
1. Get "variable" (thing w/ radical) by itself - see next question for details if multiple radicals w/ variables 2. Raise each side to the m/n power to cancel the radical (Mult by recip power) - if either m or n are even, must put abs val around var term 3. Check solution!
How do you find real roots of a polynomial by graphing?
1. Get zero on one side of equation 2. Plug it into calc 3. Find zeros using 2nd -> trace
How do you find all the roots just by knowing the polynomial?
1. Graph polynomial on calc 2. Find first rational zero from calc 3. Divide that zero into equation - keep doing this with zeros found by calc until resulting equation is of degree 2 4. Factor that equation normally 5. All divisors and final x = answers are all roots
How do you solve a quadratic/linear system by graphing?
1. Make both equations into good graphing forms - quadratic must find vertex through x = -b/2a 2. Find intersections
How do you graph a function just through knowing the zeros?
1. Plot zeros 2. Find x-val exactly in between each set 3. Plug that x-val into function to get y-val 4. Those y-vals are the turning points 5. Plot em and connect the lines 6. And find end behaviors to draw those in too
How do you find the zero for a transformed function polynomial?
1. Plug zero in for "y" 2. Solve for x
How do you solve by completing the square (no coefficient out front)?
1. Rewrite the equation into ax"+ bx = c (vars on one side, # on other) 2. Complete the square by adding (b/2)" to each side of the equation -NOTE: If equation is already in perf square trinom form, do not do these ^^^^ steps 3. Factor left side 4. Take square root of each side 5. Solve for X -REMEMBER plus and minus!
How do you synthetically divide?
1. Set divisor (second term) equal to zero and solve for var 2. Place var in box at top left and to the right write the coefficients of the equation 3. Bring down first coefficient underneath line 4. Multiply number under line by box # -then place that underneath next coefficient -add rhe coefficient and that val together 5. Repeat step 4 until no more coefficients to add by 6. Answer given are coefficients & remainder of factored equation
How do you find the inverse function of a function?
1. Switch x and y in the function 2. Solve for y (REMEMBER: if you take even root of both sides, PLUS OR MINUS!! Two diff functions result!)
When a problem asks for a polynomial, having given you some zeros, how do you do it?
1. Write factorization using zeros 2. Foil (Irrational factorizations first!!!!!!)
To make a quadratic in standard form into a quadratic in vertex form:
1. find vertex using x=-b/2a 2. plug y-int in to get "a" value 3. Ta da!
When you simplify rational things, look for _______ When you solve rational things, look for ________
1. restrictions 2. extraneous solutions
When rolling two die, what's the denom?
36
Grouping is only with
4 terms
What are the stats on a normal deck of cards?
52 cards total 26 of each color 4 suites (so 4 of each number card) 13 cards in each suite
What are the parameters for a and b in an exponential growth function?
A > 0 and b > 1
Tell me about exponential decay
A > 0 and b is in between 0 and 1 Function is going downwards to the right It decreases for all real numbers B is decay factor D: all real numbers R: y > 0 Asymptote at y = 0
What are the parameters for a and b in an exponential decay function?
A > 0 and b must be in between 0 and 1
Tell me about exponential growth
A > 0 and b> 1 Function looks like going upwards to the right Y-int: (0, a) It increases for all real numbers B is growth factor D: all real numbers R: y>0 Asymptote at y = 0
K is
A CONSTANT, so don't think of it as a variable
How do you write the exponential form?
A ^m/n^ = ^nVa^m OR (^nVa)^m
To graph reciprocal functions find
A couple points on the graph
What is a Conic section?
A curve you get by intersecting a plane and a double cone. What the intersection could be is a parabola, circle, ellipse, of hyperbola
What is the composition of inverse functions rule?
A function and its inverse follow the rule that (f^-1(f(x)))=x AND (f(f^-1(x)))=x (So both must work for a function and its inverse to be right)
If a restriction from the denominator does cancel with the numerator, what is it?
A hole (removable discontinuity)
What is an asymptote?
A line that the graph approaches but never reaches as x or y increases or decreases
What is a simulation?
A model of an event used to exist image experimental probability of an event.
What is a polynomial?
A monomial or a sum of monomials. Basically just one or multiple terms. Ex: 5x + 2
What is horizontal translation/shift?
A move of a graph horizontally (equation is counter-intuitive): F(x+h) ..... shift LEFT h units F(x-h) ..... shift RIGHT h units Subtraction or addition occurs INSIDE the parentheses
What is a vertical translation/shift?
A move of a graph up or down: F(x) + k ..... shift up k units F(x) - k ..... shift down k units The subtraction or addition occurs OUTSIDE the parentheses
What does a reflection across the y-axis look like in root functions?
A negative in front of the x INSIDE the radical
What is a complex number?
A number in the form: a + bi A --- real part Bi --- imaginary part
In between each term in an expanded binomial is
A plus sign!
What is the Fundamental Theorem of Algebra?
A polynomial of degree "n" has exactly "n" roots -# of zeros = degree NOTE: some zeros can be imaginary ans/or repeating
What is a monomial?
A real #, a variable, or a product of real #s and variables w/ whole number exponents. Basically just a single term. Ex: 5x
What is a combination?
A selection in which order does not matter. You are simply choosing a certain number out of a total.
What is a circle?
A set of all pts in a plane that are r units away from a center. R represents the radius
What is consistent system?
A system of equations with AT LEAST ONE solution
What is a reflection?
A transformation that is the graph flipped across an axis
What is a vertical stretch?
A transformation that multiplies all y-values by a factor greater than one So the graph rises and falls quicker (Kinda like a change in slope) Ex: g(x)=3x
What is a vertical compression/shrink?
A transformation that reduces all y-values (including vertex of abs val graphs if they ain't on x-axis) by a factor between 0 & 1 So the graph shrinks and rises less quickly (Kinda like a change in slope) Ex: g(x)=1/3 x Always by a fraction factor
What is a perfect square trinomial?
A trinomial of the form ax" + bx = c - (variables on one side, constant on other) MUST factor into an expression squared (ex: {x + 2}" -first and last terms are positive perfect squares
What does the a value in the general form of a reciprocal function tell you?
A val tells you the vertical stretches, shrinks, and reflections
If a restriction from the denominator doesn't cancel with the numerator, what is it?
A vertical asymptote (non-removable discontinuity)
Whats another name for an x-intercept?
A zero
When should you check for slant asymptotes?
AFTER YOU CANCEL EVERYTHING If a piece of the numerator cancels with the denominator, the degree could no longer be greater
If ^nVa^n and n is even, what is answer?
Abs Val of a
If equation is already in perfect square trinomial form, and you are trying to complete the square, no need to
Add (b/2)"
When should you find the x-intercept?
After canceling removable discontinuities
What is the domain?
All possible x vals that make function exist (not imaginary) For example, denominators of function operation can't equal 0
When writing solution to 3 var system, what order do you put them in?
Alphabetical
If a minus sign is present in a binomial you're about to expand, the expansion has
Alternating plus and minus signs in answer
A polynomial function that does not intersect the x-axis has complex roots only
Always
How should increasing and decreasing be written?
Always as inequalities with X UNLIKE range and domain with can be written in broader statements
If a negative is under and EVEN root, what will it yield?
An answer with imaginaries BUT DOES NOT HAPPEN IF ODD ROOT
What is a permutation?
An arrangement of items in a particular order
What is i?
An imaginary unit The square root of negative one
What is an inverse function?
An inverse function is a function with x and y flipped This then also changes the direction of the graph, the domain and range, and whether it is actually a function or not
What is the difference between the and v.s. Or probability equations?
And -> multiply probability of each event Or -> add probability of each event, then subtract overlap
What is the property of rational exponents when raising a two var/val term to a power? (Distributing power)
And only can be done if NO plus/minus in middle
If mutually exclusive events in and problem,
Answer is zero
What is a rational function?
Any function that can be written as a fraction
What is geometric probability based on?
Areas and area ratios, so always find areas first then compare them to get probability
What is the standard form of a polynomial function?
Arrange terms by degree in descending order
Where does the reciprocal parent function 1/x have asymptotes?
At x = 0 and y = 0
Where is the asymptote for an exponential decay function?
At y = 0
Where is the asymptote for exponential growth function?
At y = 0
What is the form of a perfect square trinomial?
Ax" + bx = c
What is Standard form?
Ax+By=C
What is b in an exponential decay function?
B is the decay factor
What is b in an exponential growth function?
B is the growth factor
What is the discriminant?
B"-4ac Tells you number of solutions of any given quadratic equation
In a solution system of equations, one equation will
Be multiplying the variables by their percentage = total amount x its percentage
When should you find the restrictions?
Before you cancel
Where is the directrix always located?
Behind the vertex of the parabola (not on the open side)
What is the name of a polynomial with two terms?
Binomial
Odd degree tells what about end behavior
Both different
Polynomial with EVEN degree and NEG leading coefficient has what end behaviors?
Both negative
Polynomial with EVEN degree and POS leading coefficient has what end behaviors?
Both positive
Even degree tells what about end behavior
Both same
What does a reflection look like that's flipped across both axis?
Both values change signs -f(-x)
How do you solve a system of inequalities?
By graphing
What are the two ways you can solve a quadratic-linear system?
By graphing or alegbraically
How can you solve systems of equations?
By graphing, by the substitution method, or by the elimination method
Rational Numbers
Can be written as fractions
Irrational Numbers
Can't be written as fraction
What is slope?
Change in Y values, over change in x values
How can you tell if a graph has vertical shrinking/compression?
Check difference between points, if it's same as original, you good
When solving absolute value equations, you should ALWAYS
Check for extraneous solutions (in the ORIGINAL equation)
What is SUUUPER important in solving rational equations?
Checking for extraneous solutions
What is the r value for a weak negative scatter graph?
Close to -0.5
What is the r value for a strong negative scatter graph?
Close to -1
What is the r value for no correlation scatter graphs?
Close to 0
What is the r value for weak positive scatter graphs?
Close to 0.5
What is the r value for strong positive scatter graphs?
Close to 1
What's the difference between using combinations/permutations and using theoretical probability rule/Fundamental counting Principle?
Combinations/permutations give outcomes Mult + div are used to analyze the outcomes
How do you add, subtract, and multiply complex numbers?
Combine like terms and treat "i" like a variable - REMEMBER that i" is -1 So foil like regular and such
To draw/describe transformations of radical function, what must you do?
Complete steps in PEMDAS order -- treat radical like parenthesis
What is the name of a polynomial with degree of 0?
Constant
If a number is a decimal, and youre trying to take a root
Convert into fraction
If you're trying to complete the square, and one variable doesn't have a b variable or value, what do you do?
Count it as zero
What is the name of a polynomial with degree of 3?
Cubic
Domain for all parabolas we know about so far
D: all real numbers
What is the domain and range for an exponential growth function?
D: all real numbers R: Y>0
What is the domain and range for an exponential decay function?
D: all real numbers and R: Y>0
How do you deal with complex fractions of rational expressions?
Deal with larger numerator and denominator as two separate addition/subtraction problems until the whole problem is just two divisions over each other
Answer given by synthetic division should always be a ______ less than original equation
Degree
The number of zeros is always the _____ of the polynomial
Degree
How does the degree of a polynomial effect the max # of turning points?
Degree has to be at least 1 more than # of turning points
Inverse relations occur in
Denominator
What do you multiply complex number fractions by?
Denominator over itself EXCEPT, denominator has opposite MIDDLE sign -unless denominator is a monomial, then it's just negative in conjugate
What are complex conjugates?
Denominator with opposite middle sign
What is a slant asymptote and how can it be found with rational functions?
Diagonal gap in graph that function gets infinitely close to but never touches SHOWED by Y= Occurs when degree of numerator is greater than the degree of the denominator. Found by synth div of restriction into numerator
What is the property of rational exponents when raising a fraction to an exponent?
Distribute exponent to numerator and denominator
When completing a square w/ a leading coefficient, what do you do with factored out leading coefficient, after equation is factored?
Divide both sides by it
If the degree of the numerator is equal to the degree of the denominator of a rational function, what do we know about the asymptotes?
Divide leading coefficients of num and denom (n/d) Quotient is Horizontal asymptote (y=)
How do you find the direct variation equation from looking at a data table?
Divide y vals by corresponding x vals If each pair gives same quotient, that is k value and direct variation is occurring
The domain for addition and subtraction operations usually
Don't change
If situation where multiple radicals with variables under them in equation, how do you solve?
Dont keep radicals on same side -- split them up Then raise to reciprocal power like normal
What is the short hand for all real numbers?
Double lined R
How do you graph hyperbolas?
Draw dotted box outline connecting the vertices to the "Co-vertices" Draw dotted diagonal lines through corner of box and center Diagonal lines are asymptotes -use function to judge vert or hort
If a graph has both negative end behaviors, the Polynomial has what degree and what sign L.C?
EVEN degree and NED leading coefficient
If a graph has both positive end behaviors, the polynomial has what degree and what sign L.C?
EVEN degree and POS L.C
What is the pattern in pascals triangles
Each number is the sum of the two directly above it Each row starts with 1 and then the row number There is always one more term than the row number
What is i,portant to remember when writing possible rational roots?
Each term could be + or -
What do no correlation graphs look like?
Either random points across the graph or a parabola because its a LINEAR correlation
How should you shift reciprocal functions?
Either shift asymptotes and keep points in same relationship to them OR Shift each point individually
IN addition to turning points, the degree of a polynomial effects the
End behavior
Middle sections of graph aren't included in
End behavior
What are horizontal asymptotes also called?
End behavior asymptotes, because that's what they effect.
The foci are ______ distance from the _______
Equal, center
How do you do increasing and decreasing with a three-part graph?
Evaluate each section individually
Where should you take restrictions from?
Every piece of the denominator that there ever was
Roots, zeros, and x-intercepts are all
Exactly the same thing (where a function crosses x axis)
What is factorial Notation?
Exclamation point "!" That indicates that you will multiply the number before it with all of the number underneath it Used only w positive numbers
What is the general form of absolute value functions?
F(x) = a |x-h| + k
How do you find the vertex in an absolute value function?
F(x) = a |x-h| + k (h, k)
How should you treat the 'a' in absolute value functions?
F(x) = a |x-h| + k A should be treated as slope
How do you find the axis of symmetry in absolute value functions?
F(x) = a |x-h| +k X=h is axis of symm
What is the general form of a radical root function?
F(x) = aVx-h +k FOR non-square root functions, just write n outside of radical Where a = vert shrink or stretch, h = horizontal shift, and k = vertical shift
What is the parent function of absolute value functions?
F(x) = |x| Looks like a V
What is the parent reciprocal function?
F(x)= 1/x Where x cannot equal 0
What is the quadratic formula?
FIND A PICTURE
If the remainder of a division problem is zero then the divisor was a
Factor
How do you add OR subtract rational expressions?
Factor Multiply each expression by whatever is needed over itself (so equiv to one) to create a common denominator between the expressions Combine numerators Combine like terms
How do you solve rational equations?
Factor Multiply each expression to get common denominator Multiple whole equation by common denominator - Everything stops being fractions Now you have equation just with numerators and easy to solve
When factoring, always
Factor out GCF first!
If completing the square and squared variable, is negative, what should you do?
Factor out negative Remember to multiple (b/2)^2 by the negative before adding to other side!
If, in an absolute value function, you have the x multiplied by something within the absolute value signs
Factor the number out
How do you multiply rational expressions?
Factor, Cancel and multiply across
How does a sum of perfect squares factor?
Factors into two binomials that are the same except for middle sign and have i with last term Ex: x" + 16 factors into (x + 4i)(x-4i)
How do you complete the square?
Find picture
What is the distance formula?
Find picture
What is the standard form equation of a circle?
Find picture Where (h,k) is the center
What is the general form of reciprocal functions?
Find picture y= a/(x-h) + k
How do you find the axis of symmetry & vertex from the intercept form of a quadratic function?
Find points by just setting x sections equal to zero. Then point directly between them is axis of symmetry Axis of symmetry gives you x coord of vertex -plug that in for the y coord
How do you solve those pesky SAT questions?
Find rate for each thing as one unit in smallest amt of time add in equation to get goal thing in smallest amt of time solve
How do you use differences to find degree of a polynomial?
Find the differences between the y-vals of a function Keep going between the differences until they all have the same difference The number of times you found the difference is the degree of the polynomial
How do you solve a system of equation by graphing?
First make into slope intercept form Then graph different lines Intercept is solution
What is the sum of cubes formula?
First sign is the same, second is diff
How do you write out the expanded variables along with the coefficients using pascals Triangle?
First variable starts on left and moves right, decreasing in degree (starts as degree of expansion, goes down to 0 -> which is just 1)
How do you divide rational expressions?
Flip second one into reciprocal and then follow multiplication process - factor, cancel, multiply across
Closure Property
For addition: a+b is a real # For multiplication: axb is a real #
What is the (n+1) Point Principal?
For any set of n+1 points in the coordinate plane that pass the vertical line test, there is a unique polynomial of degree at most n that fits the points perfectly
What is the exponential form?
Form of phrase that indicates the principal root Specifically, the exponential form is the reciprocal of the root (often as fraction. Recip of root but also power of val) Ex: square root -> 1/2 power
How do you complete the square?
Get constants on one side Group like terms together Factor out any GCFs Find (b/2)^2 Add to left side Multiply by GCF and add to right side
How do you solve a three var system of equations by elimination?
Get the coefficients of one variable to match so that they will cancel -may require multiplying one of the equations With your new two var system of equation, solve them for one var Plug that var back into your two var system, for second var Plug those two vars back into an original equation
What's essential to addition/subtraction of rational expressions problems?
Getting common denominators
What is the result of polynomial long division?
Gives you factored version of equation if divisor was factored out
If a linear/quadratic system has no solutions, line
Goes completely beneath quadratic
If a linear/quadratic system has two solutions, line
Goes through quadratic twice
Polynomial with odd degree and pos leading coefficient has what end behaviors?
Going left (towards neg infinity), negative (going down)_ Going right (towards pos infinity), positive (going up)
Polynomial with odd degree and neg leading coefficient has what end behaviors?
Going left (towards neg infinity), positive (going up) Going right (towards pos infinity), negative (going down)
Polynomial has negative leading coefficient, what do you know about end behavior?
Going towards the right (pos infinity) the graph will be negative (going down)
Polynomial has positive leading coefficient, what do you know about end behavior?
Going towards the right (pos infinity) the graph will be positive (going up)
If a zero has an odd multiplicity, what does that tell you about the line behavior?
Graph crosses x-axis at that zero
If a zero has an even multiplicity, what does that tell you about the line behavior?
Graph does not cross x-axis, just touches at that zero and turn parabolically
How do you solve a systems of INEQUALITIES by graphing?
Graph each line Shade Overlap is all solutions
How do you graph absolute value inequality functions?
Graph like normal absolute value function but then: -draw dotted line if not equals to -draw solid line if also equals to then shade: -outside if less then -inside if greater than
If you can't factor a phrase to find the zeros, what should you do?
Graph phrase on calc and find x-intercepts
In solving inequalities, you end up with an
GreatOR solution or less thAND solution
Square root function's domain and ranges always have
Greater/less than OR EQUAL TO
What does the h value in the general form of a reciprocal function tell you?
H val tells you the horizontal shift WHICH changes where the asymptotes are, not the vertexes
What is crucial to remember in graphing?
HOY VUX
What is an independent system?
Has ONE solution -diff lines -diff slopes
What is a dependent system?
Has infinitely many solutions -the same line (same slope, same int)
What is an inconsistent system?
Has no solution -parallel lines -same slope -diff int
If degree of numerator is less than the degree of the denominator of a rational function, what do we know about the asymptotes?
Horizontal asymptote at y=0
End behavior is always
Horizontal asymptote. IF SLANT ASYMPTOTE, end behavior is infinity and negative infinity correspondingly
What are horizontal asymptotes and how do you find them with rational functions?
Horizontal gaps in graph that function gets infinitely close to but never touches. SHOWS up as Y= Found by comparing degree of numerator to degree of denominator
What is the multiplicity of a zero?
How many times that zero is a factor of the function
What is the end behavior of a polynomial?
How the graph acts to the far left and far right
How do you solve the 6.8 radical equations by algebra?
I don't know, ask MR. Naydan
How must you write quotients/products/sums of complex numbers?
IN STANDARD FORM A + bi
What are the A, B, and C in standard form?
INTEGER coefficients (whole numbers including 0 and negatives, although try to make them positive)
What is the theoretical probability equation?
If a sample space has n equally likely outcomes, then the theoretical probability is: P(A) = m/n Where m is how many time you get what you want And n is the number of equally likely outcomes
What is the nth root Principle?
If a^n =b with a&b as real #s and n as a positive integer, than a is the nth root of b ex: 2 squared = 4 then 2 is the square root of 4
What rule can be used with Horizontal Line Test?
If any horizontal line drawn on the graph will hit the graph in more that's one sport that's the function's inverse will NOT be a function. Basically, the horizontal line test for a function is the vertical line test for it's inverse
In what situations of m and n will you put abs val around the simplified variable term?
If either m or n are even WHEN YOU ARE MULTIPLYING both sides to the m/n power to cancel root(S)
What is the Fundamental Counting Principle?
If event M can occur in m ways and is followdd by event N, ehich can occur in n ways, then the event M followed by event N can occur in m * n ways
When do you put absolute value around answer?
If original val is even rooted, but simplified version is to odd exponent (including 1)
What is the Vertex Principle of Linear Programming?
If there is a max or min value of the linear objective function, it occurs at 1 or more vertices of the feasible region (max & min are in corners of feasible region)
What is the rule for putting exponents on the inside or outside of radicals?
If whatever is under the root is already a perfect of that root, put exponent on outside
What is the Remainder Theorem?
If you divide the polynomial P(x) by x-a then the remainder is P(a)
If a linear/quadratic system has one solution, line
Is tangent to quadratic -skims bottom or hits bottom corner
If a graph does not cross x-axis but does touch it, what does that tell you about that zero?
It has even multiplicity
If a graph crosses an x axis, what does it tell you about that zero?
It has odd multiplicity
How do you find the axis of symmetry of a quadratic in vertex form?
It is X = h
An absolute value inequality is all real numbers if
It is greater than a negative
What does the equation b"-4ac tell you?
It is the discriminant Tells you number of solutions of any given quadratic equation
How do you find the maximum of a downward facing parabola?
It is the y-val of the vertex
How do you find the minimum of an upward facing parabola?
It is the y-val of the vertex
When there is a negative sign in front of the x" of a quadratic function, that means
It opens downwards (is reflected)
When there is no negative sign in front of the x" of a quadratic function, that means
It opens upwards
What does the equation (b/2)" tell you?
It tells you what to add on either side of equation during completing the square process. - See, completing square w/ leading coefficient process for examples of exception
What does a hollow circle in function operations mean?
It's just the equivalent of another set of parentheses Ex: (f o g) (x) equals f(g(x))
If you use the conjugate of a binomial, what happens?
It's just the first and last term squared, (last term is usually negative)
If ^nVa^n and n is odd, what is answer?
Just a
What does a reflection across the x-axis look like in other root functions?
Just a negative OUTSIDE the radical
To divide something by a fraction
Just multiply the something by the reciprocal of the fraction instead
How do you multiply binomial Radical Expressions?
Just use FOIL
What does the k value in the general form of reciprocal functions tell you?
K value tells you the vertical shift WHICH changes where the asymptotes are, not the vertexes
In an ellipse, a is always
Larger than b
If mutually exclusive events in or problem,
Leave off subtraction
The increasing and decreasing is read
Left to right
Smaller the sample size of simulations
Less accurate the answers
How do you write end behavior
Lim f(x)= ____. X -> -infinity X-> + infinity "As x is towards neg infinity" Going left. Lim f(x) = ________ X-> + infinity "As X is towards pos infinity" Going right
What is the name of a polynomial with degree of 1?
Linear
How do you find the domains of function operations?
Look at where equation is factored but not canceled/ or original equation and ending equation LOOK FOR all possible x vals that make function exist and write down what x cant be for function to be non-imaginary
What does a reflection across the x-axis look like in a square root function?
Looks like a negative outside square root
How do you solve a problem where they give you minimal information about the polynomial and a couple example points?
Make a systems of equations
When graphing quadratics, you may need to
Make a table for other points (besides vertex and y-int)
How do you solve a linear-quadratic system algebraically?
Make both y= and then substitute OR Eliminate and then take square roots REMEMBER to plug back in for y!
To find "a" of a quadratic equation
Make into vertex form Plug in other point
How do you solve a quadratic system of inequalities?
Make into y= Find vertexes Graph with dotted or solid lines Shade Overlap is solutions REMEMBER make table if need be for other points!
How do find the LCM of two polynomials?
Make new fraction that has basically all of the two formulas - If anything is unique to one side, it's carried to the new equation - If something is on both sides of the equation, take the larger amount of it - If any constants, do prime factorization before - If polynomials with variables, factor first
What is the name of a polynomial with one term?
Monomial
How do you cancel equations with root on one side?
Mult both sides by m/n power
If asked to find a certain degree function with less than that no. Of real zeros, think
Multiplicities of those zeros or imaginary zeros
How do you find the inverse variation equation from looking at a data table
Multiply X and Y values together If each pair gives same product, that is K value and inverse variation is occurring
Inverse Property of multiplication
Multiply a number by its reciprocal and you get 1
How do you rationalize the denominator if it is a radical binomial?
Multiply by conjugate over itself
What is the property of rational exponents when raising a power to another power?
Multiply exponents
If the variables in the binomial you're expanding have coefficients already, what do you do in the expansion
Multiply the Pascal's triangle numbers by the coefficients
How do you rationalize a cube root fraction?
Multiply the fraction by anything under the Radical that will make the demoninator a perfect square - top gets multiplied under Radical - bottom gets multiplied to perfect cube and then simplifies down to the cube root of the number
How do you rationalize a fraction with a radical in the denominator?
Multiply whole fraction by radical denominator over itself - denominator becomes itself w/o radical - numerator becomes itself times denominator radical
How do you graph vertical stretch/shrinks for radical functions?
Multiply y-évalués by factore
What are the number limits on a probability?
Must be between zero and one
How do you do range if big gap in graph (ex: three part graph)?
Must use inequality
What's a tip for assigning N and R values in number of permutations?
N is always larger than R
If the remainder of a division problem is not zero, then the divisor was
NOT a factor
Whole Numbers
Natural numbers (counting numbers) but with 0
Polynomial functions with complex coefficients only have one complex zero
Never
How do you write the answer of polynomial long divisiom
No remainder- just write quotient as answer Remainder - write quotient + remainder/divisor
Can graphs ever cross vertical asymptotes?
No, never
When doing function operations, when should you factor further ?
Not in addition, subtraction, or multiplication, cuz you're just trying to get new equation BUT in DIVISION factor/simplify fully
For a hyperbola, what matters in the equation?
Not the denominator but the prder
What is the index?
Number of root of radical
Joint variation occurs in
Numerator w/ more than one variable next to k
Direct variation occurs in
Numerator with just k and ONE variable
An absolute value can never be equal to or less than a
Négative number
How do you combine Radical Expressions?
ONLY possible if the index and radicand are same Treat the Radical like a variable. Add the "coefficients" and keep one unit of radicand (don't multiply or do anything under the radical)
Identity Property
Of addition: a + 0 = a Of multiplication: a x 1 = a
How do you rewrite something in the standard form?
Often times complete the square
The general form of radical functions looks exactly same as absolute value functions, etc...
Okay
When completing a square with a leading coefficient, what do you add to both sides?
On left side you add (b/2)" AND on right, you add (b/2)" but multiplied by leading factored coefficient -REMEMBER THAT B IS AFTER FACTORING VERSION
Where is all the other information for this test?
On the test itself
Where are the foci of a hyperbola located?
On transverse axis
Anything to zero power is equal to
One
For so,e of the Conic Sections, they must be set equal to
One
If there is no vertical stretch in a reciprocal function, where is the first point?
One away from each asymptote
How do you graph the absolute value of a complex number?
One axis is imaginary (draw the imaginary part of the comp #) Other axis is real (draw real part of comp #) Should be two perpendicular line Draw right triangle from these - hypotenuse is absolute val of complex number
What is an exponential function?
One following the form FIND PICTURE Where a does not equal 0 and b is greater than one and doesnt equal 1
When should you graph a function?
Only after you've canceled everything.
What are conjugates?
Opposite middle sign
What is the conjugate axis of a hyperbola?
Opposite of transverse axis--gives you width of the hyperbola B values
Can factorials be canceled?
Oui Oui!
What are equally likely outcomes?
Outcomes when each outcome in a sample space has an equal chance of occurring
When you multiply by conjugate, which terms cancel out?
Outer and inner
What is the equation for Probability of A and B?
P(A and B) = P(A) * P(B)
What is the equation for Probability of A or B?
P(A or B) = P(A) + P(B) - P(A and B-- The overlap) IF THE EVENTS ARE mutually exclusive, ignore the subtraction Since there is no overlap
What is the experimental probability equation?
P(event) = (# of times the event occurs)/(# of trials) In DECIMALS
If taking even roots, REMEMBER
PLUS OR MINUS
What do you have to remember in synthetic division and polynomial long divisiom
Place holders! All day e'ry day
If a problem just gives you "a" then
Plug "a" into equation as "x" val and solve (because x-a=0, x=a)
How do you find the Y intercept?
Plug 0 in for X then solve for Y
How do you find the X intercept?
Plug 0 in for Y then solve for X
How do you solve the 6.8 radical equations by graphing?
Plug equation given into calc as function (1 graph) Also plug value given into calc as other function Find intersection
To find other points on parabola
Plug in other inputs
In a linear programming problem, what do you do once you find the vertices of the feasible region?
Plug them into the objective function Largest no. is point that maximizes function Smallest no. is point that minimizes function
If you see f(g(x)) and the x is a number, what should you do?
Plug x into g(x) first then plug that end value into f(x)
What is the name of a polynomial with four or more terms?
Polynomial with ___ terms
What's useful to use when simplifying?
Prime factorization tree
What are decomposition problems?
Problems reverse of composition--you just find what was plugged in
What is the even root of a neg?
Pull out i and then factor
If either m or n are even when multiplying radical equations to the m/n power to cancel out root(s), what do you need to do?
Put abs vals around simplified no radical variable term.
Where is a reciprocal function without a negative sign graphed?
Quadrants 1 & 3 IN RELATION TO ASYMPTOTES
Where is a reciprocal function with a negative sign graphed?
Quadrants 2 & 4 IN RELATION TO ASYMPTOTES
What is the name of a polynomial with degree of 2?
Quadratic
What is the name of a polynomial with degree of 4?
Quartic
What is the name of a polynomial with degree of 5?
Quintic
What is the equation for rate?
R=d/t
In some systems word problems, you must find
Rate first
Imaginary zeros are not
Rational
Last Val given by synthetic division is the
Remainder
Parabolas are
SYMMETRICAL
What does "like Radicals" mean?
Same index and radicand
What is the conjugate of a binomial?
Same numbers, different middle sign
How do you write a transformed polynomial function?
Same way as did with quadratics F(x)=(ref)(stretch/compress)(x +/- horz shift)^degree +/- vert translation
Imaginary zeros won't
Show up on calc. THIS is why you need to synthetically divide to factor.
Before doing any work, what should you always do?
Simplify
When rough sketching a rational function, what should you do first?
Sketch asymptotes and holes first and then find specific points on calculator
After you multiple both sides by m/n power, what is your objective?
Solve for x
How do you solve a three var system of equations by substitution?
Solve one equation for one of the variables Plug that into the other two equations Use the process of elimination with those two Plug answer into one of the two new equations Plug those two variables back into an original equation
A polynomial function with real coefficients has real zeros
Sometimes
How do you solve an absolute value equation?
Split into two parts where val on other side of equation from variable is positive in one situation and negative in another
What is important to remember near the end when using the quadratic formula?
Split the parts on either side of the +- into two separate fractions for simplifying
To write answer given by synthetic division as equation
Start from left, ignore first val, assign next val to "x", next to "x"" etc...
Parent function f(x) = x"' looks like
Starts down on left side, goes up, horizontal through (0,0), then turns upwards again
If polynomial looks huge and unmanageable to factor, use
Synthetic division
What do you need to remember when writing transformed polynomial function?
THE DEGREE GOES OUTSIDE THE PARENTHESES
If there is a square root of a negative number,
Take i (thus the neg) out. Left with iV (number w/o neg)
If a number is a fraction and you're trying to take a root
Take root of numerator and denominator individually
To reverse a number with the exponent,
Take the exponent root of the result
How do you take the square root of a fraction?
Take the square roots of numerator and denominator individually -if denominator has radical after, RATIONALIZE
What is the Horizontal Line Test/what does it tell you?
Tells whether a function's inverse is a function or not
If there are two vertical asymptotes and they aren't equal distances away from the y-axis, what can you assume?
That the min/max of the parabola in the center of the graph is not directly on the y axis
When counting number of ways you can get a certain sum from a roll of two die, what do you have to take into account?
That the order you roll the numbers on the different die matters Ex: 2 & 3 is diff than 3 & 2 and they must be equally counted
If the "a" is negative, the parabola opens downwards, which indicates
That there is a MAX value (highest point), which is the y-coordinate of the vertex
If the "a" is positive, the parabola opens upwards, which indicates
That there is a MIN value (lowest point), which is the y-coordinate of the vertex
How does Pascal's Triangle correlate to the binomial expansion?
The Numbers of the Triangle are the coefficients in the expansion. And the row number is the degree of expansion.
What is the Rational Root Theorem?
The POSSIBLE RATIONAL roots are found by taking all the factors of the CONSTANT term OVER the factors of the LEADING coefficient
Exponential form is reciprocal of ____ _____ AND the _______ of the ______
The Root and power of the value
In linear functions, a vertical shift and a horizontal shift are
The SAME thing
When completing a square with a leading coefficient, remember that the "b" in the equation (b/2)" is
The b from the factored version of the equation, not the b from before the leading coefficient was factored out
What is the focal length?
The distance between the vertex and focus
What are the domains and ranges of an inverse function compared to in the original function?
The domain of original gets flipped to be range of inverse and the range of original gets flipped to be domain of inverse
What are the vertices of an ellipse?
The end points of the major axis Picture
What are the co-vertices of an ellipse?
The endpoint of the minor axis Picture
When graphing a scatter plot on the calculator, write down
The equation for the line of best fit, and the correlation value. Round the equations constants to 2 decimal places
What does the equation x=-b/2a help you with a quadratic in standard form?
The equation gives you the axis of symmetry, and thus, the x coordinate of the vertex -then to get y coordinate of vertex, plug X back in (y coord tells you max/min val)
What is the degree of a monomial?
The exponent of the variable NOTE: degree of "x" is 1 and degree of a whole number is 0
What is the leading coefficient of a polynomial?
The first coefficient--the coefficient of the highest degree term (when polynomial is written in standard form)
What is an objective function?
The function for which you are trying to maximize or minimize a quantity
What is the degree of a polynomial?
The greatest degree among its monomial terms
Associatives Property
The grouping of addition or multiplication does not matter
What is the end behavior for reciprocal functions?
The horizontal asymptote
The solutions of quadratic systems are
The intersections, not the "zeros"
What is the LCM?
The least common multiple -- smallest thing that both can multiply to
What is the m and what is the n in x^m/n?
The m is the power that in the radical form of the equation, x was raised to The n is the power of the root that in the radical form of the equation, x was under
What is linear programming?
The method for finding a max or min val given a set of constraints
What is this center of a hyperbola?
The midpoint between the vertices
What is the number of combinations equation?
The number of combination ps of n items of a set arranged r items at a time is: nCr = n!/r!(n-r)!
What is the equation for number of permutations?
The number of permutations of n items of a set arranged r items at a time is: nPr = n!/(n-r)! N must be larger than R
What is the simplified understanding of the (n+1) point principle?
The number of points a function gives you minus 1 is the highest possible degree of the polynomial that models the function
If the denominator of a complex number fraction is a monomial, what do you multiply the fraction by?
The opposite sign denominator over itself (w/ opposite sign)
Commutative Property
The order of addition or multiplication does not matter
In a problem with multiple translations, you complete them in
The order of operations
If "h" is subtracted outside the parenthesis of a quadratic in vertex form, what transformation has occurred?
The parabola is shifted DOWN h units -subtract h outside parentheses means shift down
If "h" is added inside the parenthesis of a quadratic in vertex form, what transformation has occurred?
The parabola is shifted LEFT h units -add h inside parentheses means shift left
If "h" is subtracted inside the parenthesis of a quadratic in vertex form, what transformation has occurred?
The parabola is shifted RIGHT h units -subtract h inside parentheses means shift right
If "h" is added outside the parenthesis of a quadratic in vertex form, what transformation has occurred?
The parabola is shifted UP h units -add h outside parentheses means shift up
When "a" of a quadratic function is between zero and one, what does that mean about the parabola?
The parabola is vertically compressed/shrunk by a factor of "a" It also becomes wider
When the "a" of a quadratic function is greater than one, what does that mean about the parabola?
The parabola is vertically stretched by the factor of "a" It also becomes narrower
What are the turning points of a graph?
The places where a graph changes in direction (relative max or min)
If b"-4ac < 0 (discriminant)
The quadratic has 0 REAL solutions - or 2 imaginary solutions (doesn't matter as much for right now) IGNORE PIC ON THIS 0 real solutions!!!!!!!
If b"-4ac = 0 (discriminant)
The quadratic has 1 REAL solution
If b"-4ac > 0 (discriminant)
The quadratic has 2 REAL solutions
What is a rational expression?
The quotient of two polynomials
What is a feasible region?
The region that contains all the points that satisfy the constraints
What is the transverse axis in an hyperbola?
The segment connecting the vertexes A values
What is the minor axis of an ellipse?
The segment perpendicular to the major axis at the center, and it is the shorter axis Picture
What is the major axis of an ellipse?
The segment that contains the Fock and has its endpoints on the ellipse. It's midpoint is the center of he ellipse and it is the larger axis. Find picture
What is a sample space?
The set of all possible outcomes to an expriment
What is a parabola?
The set of all pts in a plane that are the same distance from a fixed line (directrix) and a fixed pt not on the line (focus)
What is an ellipse?
The set of all pts in a plane where the sum of the distances from 2 fixed points (foci) is a constant number
What is a hyperbola?
The set of points in a plane where the difference between the distances from 2 fixed pts (foci) is constant. Looks like two parabolic shapes
What is a parent function?
The simplest form in a set of functions that form a family
What is the "correlation" in a scatter graph?
The strength of the trend
In an "after how many" systems word problem
The two variables will be something like price and hours, not assigned to individual objects
The distance from the vertex to the focus of a parabola is the same as the distance between
The vertex and the directrix
Tell me about vertexes of hyperbolas?
The vertex of each parabolic part is counted as a vertex of the hyperbola
What is the focal width?
The width of the parabola through the focus Find picture
How are the points of a inverse function in relation to the points of the original function?
The x and y coordinates are switched
What does the equation x = -b/2a tell you?
The x coord of vertex and/or the axis of symmetry
In the graph of an inverse function, all the points are reflected across
The y = x axis
If, a^n=b, and n is even AND b is positive
There are 2 real nth roots of b: Positive root (Principal Root) = ^nVb (nth root of b) Negative root (Neg) = ^-nVb (neg nth root of b)
When a system gives you a true statement,
There are INFINITELY MANY SOLUTIONS
If, a^n=b,and n is even AND b is negative
There are no real nth roots/two imaginary roots ex: a negative number can't be a perfect square/even exponent
Just because an equation isn't factorable, doesn't meant
There are no solutions
If there are two vertical asymptotes, what does that tell you about the graph?
There are probably three separate graph portions
When a system gives you a false statement
There is NO SOLUTION
If the degree of the numerator is greater than the degree of the denominator of a rational function, what do we know about the asymptotes?
There is a slant asymptote and no horizontal asymptote. Found by dividing restriction into numerator by synthetic division. For answer, first number is x coef, second is the constant, third can be ignored y=
If, a^n = b, and n is odd
There is one real nth root of b, written as ^aVb The ath root of b
What do perpendicular equations look like?
They have opposite reciprocal slopes and different y intercepts
What do parallel equations look like?
They have the same slope but different y intercepts
How do you deal with negative exponents?
They make term Switch POSItions (if on bottom, moves to top and vice versa) Turn exponent POSItive (no longer neg exponent, but still present)
For linear functions, what do vertical compressions do?
They make them flatter
For linear functions, what do vertical stretches do?
They make them steeper
For parabolas, what do vertical stretches do?
They make them thinner
For parabolas, what do vertical compressions do?
They make them wider
What is the qualification to do a probability of A AND B?
They must be independent events
What is the property of rational exponents when raising a var/val to a negative exponent?
Thing gets flipped and exponent becomes positive
What is the radicand?
Thing under radical
What are restrictions?
Things that variables in the denominator can't equal so that equation is undefined
What is important to remember when finding variables in a three var systems of equations?
To get first var, solve the two var system you made To get second var, plug first var into TWO VAR SYSTEM To get the 3rd var, put 1st & 2nd vars into an ORIGINAL eq
If you see (f o g)(x) or f(g(x)), what does it mean?
To take the thing on the right (here g(x)) and plug it in as x in f(x)
What is the name of a polynomial with three terms?
Trinomial
The degree of a polynomial effects the maximum # of
Turning points
If a graph is constantly increasing, tell me about turning points and increasing and decreasing
Turning points: None Increasing: all real #s Decreasing: Never Cuz it's just increasing at different rates even though it might look flat
What are mutually exclusive events?
Two events that cannot happen at the same time. If or problem with mutually exclusive events, leave out subtraction
How do you use a simulation?
Use RAND INT on calculator Input three numbers (lowest possible, highest possible, how many random integers) Run at least 20 simulations Count the number of simulations in which a certain number appears __ or more times Number of sims that the abundant certain number occurred /OVER the total sims gives you experimental probability of event!
How do you solve a problem using the (n+1) point principle?
Use calc to graph scatter plot Use principle to find highest possible degree Use calc and points to find function of that degree that fits line perfectly (r should equal 1)
How do you get an integer answer when multiply binomials?
Use conjugate
What is the absolute value of a complex number (a + bi)?
Va" + b" Square root of a" + b" SO FORGET ABOUT THE I WHEN WRITING ABS VALUE
How do you find the vertex of a quadratic from the vertex form?
Vertex: (h,k)
What are vertical asymptotes and how do you find them with rational functions?
Vertical asymptotes are found as zeros of the denominator that do not cancel with the numerator. AKA non-removable discontinuities SHOW up as X= And they're basically gaps in the graph that the function gets infinitely close to without touching
Think through cheating
Way to factor
What do multiplicities of zeros tell you about the graph?
What the line behavior at the zero is Even multiplicity --> graph does not cross x-axis, just touches and turns parabolically Odd multiplicity --> graph cuts through x-axis at that zero
What is a removable discontinuity?
When a variable part on the numerator and denominator cancel out. This shows up as hole in graph. Is the "restriction" that cancels out with numerator
What are independent events?
When an occurence of one event does not affect how a 2nd event occurs
What are dependent events?
When an occurrence of one event affects how a second event occurs
When do you use Pascal's Triangle?
When expanding a binomial (go to level that corresponds to degree of expansiom
What is joint variation?
When one quantity varies DIRECTLY (on numerator) with two or more quantités. Kind of combined variation
What is combined variation?
When one quantity varies with respect to two or more quantities.
When do you put absolute value?
When taking even root but outcome is to odd exponent
What is a non-removable discontinuity?
When the domain restriction of a function does not cancel in the equation. This shows up as vertical asymptote Is the "restriction" that does not cancel with numerator
What is a sum of perfect squares?
When you have a binomial that is a perfect square plus another perfect square -must by factored into complex numbers Ex: x" + 16
Standard form of a quadratic function
Where a does not equal zero
Degree of polynomial tells you what about the end behavior
Whether both sides are same or not Even degree --> same Odd degree --> diff
Sign of L.C. Of polynomial tells what about end behaviors
Whether it is positive or negative going towards pos infinity Pos leading coef --> pos going towards pos infinity Neg leading coef --> Neg going towards pos infinity
Integers
Whole numbers including negatives
How do you divide complex numbers?
Write as fraction (so just move it vertically) Rationalize denominators using complex conjugates - denominator with different middle sign - in cases where denominator is monomial, write complex conjugate as negative IN reduced complex number that results, write in standard form (a + bi)
What is the factorization version of roots?
Write each root as it would be before solved for x (when set equal to zero) put parenthesis around each phrase
In an ellipse, if a is under
X -- horizontal ellipse Y -- vertical ellipse
The format for increasing and decreasing is
X < _______ X > _______ NO EQUAL TO
What are the three similar equations to remember this chapter?
X = -b/2a >> x coord of vertex of quadratic and/or axis of symm B"-4ac >> discriminant, # of solutions to quadratic (B/2)" >> added to both sides when completing the square
What are x intercepts and how do you find them with rational functions?
X intercepts are where graph crosses x axis and they are found as ZEROS of the NUMERATOR NOTE: if a part of the numerator is a removable discontinuity, it is not an x-intercept!
What is the Factor Theorem?
X-a is a factor of f(x) if and only if "a" is a zero of f(x) Basically, if there's a zero of function, the opposite of it, attached to x is a factor of the function
What does a reflection look like that's flipped across the y-axis?
X-values change signs while the y-values stay the same F(-x) So put a negative wherever you see an x
What is the y intercept for an exponential growth function (y = ab^x)?
Y int is (0, a)
What is the y-intercept for an exponential decay function?
Y int is (0, a)
What are y intercepts and how do you find them with rational functions?
Y intercepts are where the graph crosses the y axis and they are found by simply plugging ZERO in for X
What is point-slope form?
Y-Y1 = m(X-X1)
What does a reflection look like that's flipped across the x-axis?
Y-values change signs while the x-values stay the same -f(x) So multiply a negative through the function
What is a power function?
Y= a * x^b Where b is the power And a is the constant of proportionality (think direct variation)
What does a just inverse variation equation look like?
Y= k/x Where k is the constant of variatoon
What is the parent function for linear functions?
Y=X
What is the intercept form of a quadratic function?
Y=a(x-p)(x-q)
What does a just direct variation equation look like?
Y=kx Where k is the constant of variation
What is slope-intercept form?
Y=mx+b
What is the parent function for quadratic functions?
Y=x^2
Do non-removable discontinuities still count as points of discontinuity, even though they aren't holes?
Yes
Can graphs ever cross horizontal asymptotes?
Yes, but the horizontal asymptotes are firm at the ends of the graph, so the function can only cross the asymptote in the middle
How do you know if a graph crosses the horizontal asymptote and how do you find the inc/dec + range then?
You know if the x intercept(S) of the graph are ON the horizontal asymptote OR PASSED it THEN (since mini parabola), you would need to use max/min on calc for that section to help with inc/dec and range Also there would then be a relative max and min too
Explain strategy/process of grouping
You're trying to get same leftovers from both 1. Split polynomial w/ 4 terms into 2 halves 2. Factor out GCFs from both sides so that you end up with same thing in parenthesis 3. Clump GCFs together in parenthesis and write the leftover together in parenthesis 4. TA DA it's factored
What is the b in an exponential function?
Your base, the rate at which something is increasing (must be greater than zero and can't equal one)
What is a in an exponential function?
Your initial value (and cant equal zero)
Remember to distribute
Your minus signs!
What is the nth Roots of nth powers Principle?
^nVA^n Ns cancel out and if N is odd: the answer is just a N is even: answer is absolute value of a
What are the two perfect square trinomial form shortcuts when factoring?
a" + 2ab + b" = (a+b)" a" - 2 ab + b" = (a-b)"
Do difference of squares factoring
a"-b" = (a+b)(a-b)
In a word problem like How high is the bridge, or somethin, remember
add height of parabola to distance below
What is the Conjugate Root Theorem?
basically imaginary zeros always show up in pairs as do ones with radicals--all irrational zeros if a+Vb is a root of a polynomial, so is a-Vb If a+bi is a root of a polynomial, so is a-bi
A greater than/less than sign indicates a ______ line
dotted
In problems like "find equation of parabola through these three points"
establish system of equations with y=ax"+bx+c for each point. Then solve
What form do the parent functions of all roots follow?
f(x) = ^nVx
What is the inverse function notation?
f^-1(x)
You should always
factor first
What is the square root of negative one?
i
What is the imaginary unit?
i OR V-1
What is the property of rational exponents when multiplying terms?
if terms are same, exponent added
All odd root functions can go into
negatives and positives on graph
All even root functions can't go into
negatives on graph, unless transformations change domain to be possibly negative
A RELATIVE min/max is just in
one portion of the graph, not whole things necessarily
A greater than or equal/less than or equal to sign indicates a ____ line
solid
Across standard and vertex forms, the "a" value
stays the same
What is the property of rational exponents when dividing val/vars raised to powers?
subtract exponents
End behavior is only at
the ends, not in the middle
When graphing a scatter plot with hours as a value
use army time
Twice as many x as y means
x = 2y
What is THE MOST IMPORTANT equation for standard form quadratic functions?
x= -b/2a (tells the axis of symm)
Range for all parabolas we know about so far
y is less than/greater than the min/max ex: min of -2 range: y>- -2
Do phrases of removable discontinuities still count in the restrictions?
yes!!!
