Equations to know
vertex form & vertex
y=a(x-h)^2+k vertex: (h,k)
intercept form and vertex
y=a(x-r1)(X-r2), r=roots vertex: between x-intercepts x= ((x1+x2)/2) y= plug in to equation
standard form & vertex
y=ax^2+bx+c vertex: x=(-b/2a) y=plug in x
Gauss-Jordan elimination
A method of solving a linear system of equations. This is done by transforming the system's augmented matrix into reduced row-echelon form by means of row operations.
a X b = (a cross b)
-b x a
log1 or ln1 =? log10
0 1
identity matrix
A square matrix with ones (1s) along the main diagonal, from the upper left element to the lower right element, and zeros (0s) everywhere else.
nontrivial solution and linear independent or dependent?
a nonzero solution of a homogeneous equation or system of homogeneous equations linear dependent
difference of two squares
a polynomial that can be factored as the product of the sum and difference of two terms
negative exponent
an exponent less than zero which causes the base and its exponent to move positions in a fraction
how to find the area from two vectors that form a parallelogram? state formula and method
area = ||u x v|| (magnitude of the vector u cross vector v) ex. given A(0,1,3) B(1,3,5) D(4,5,3) draw out vectors & determine which two vectors you need (AB and AD) to find AB, minus B points from A and to find AD minus D points from A. so AB= (1,2,2) AD=(4,4,0) now use equation, area = ||u x v|| find the cross product of AB & AD then the magnitude (sq.rt. of the points squared)
f(x)=(2/3)^x whats the inverse?
f(x) is an exponential so its inverse is a log function, f^-1(x)= long base (2/3) of x
be careful when you see find the slant asymptote for f(x)=(x^3-1)/x this can be rewritten, think about it!
f(x)= (x^3/x)-(1/x) =x^2-(1/x) this has an asymptote of y=x
solving system of equations what can equal a point or plane?
having unique solutions 3 plans can all intersect at one point and 3 lines can as well
polar form vectors with imaginary numbers
r(cos (angle)+ i sin(angle)) where r= radius angle from the x axis going counter clock wise use 30-60-90 or a^2+b^2=c^2 to get r!
how are complex vectors written?
real number then complex (with i)!!! always z=2-4i not z=2i-4
Slope of a line
rise over run = y2 - y1 / x2-x1
vertical asymptote
set denominator equal to zero
How to find slope of complex vectors (z=2-4i)
slope = complex side/real number so -4/2= -2
If 2 vectors are perpendicular
their dot product will be 0
a x (b x c)=
(a * c)b-(a * b)c (a dot c)
a*(b x c) = a dot (b cross c) =
(a x b) * c
Unit Circle: degrees to radians (Q4) 330 315 300
11pi/6 7pi/4 5pi/3
let A and B be n X n matrices. Which of the following does not always equal (A+B)^2? a. (B+A)^2 b. A^2+2AB+B^2 c. A^2+AB+BA+B^2 d. (A+B)A + (A+B)B
*remember matrices are not commutative in multiplication and the (A+B)^2 = A^2+AB+BA+B^2 so c. is correct a. is also correct because matrices are commutative in addition d. is correct because matrices follow distribution prop so B. is the right answer because AB does not always equal BA so cant combine them to make 2AB!
Matrix Inverse when det(A)=0
- There is NO inverse - There is either an infinite # of solutions or no solution - Linear dependent
Matrix Inverse when det(A) does not = 0
- There is an inverse matrix - There is a unique solutions - Linear independent
solving system of equations what can equal a line or plane? (how many plans would you need to get infinite number solutions all the time?
- a system of equations that have an infinite number of solutions! - 2 planes that intersect would have infinite # solutions (can also be three planes - a system of equations with infinite number of solutions has all the equations graphed on top of each other in a continuous line
Synthetic division
- first find the determinate to see if invertible - create a matrix with dot line with identity matrix on the other side of the dotted line (ones and zeros) solve with row operations to get the right side to look like the identity matrix and left will be your inverse of that original matrix!
Log base x (x) or log base5 *5
1
b^2-4ac = 0
1 real solution
cos^2 = sin^2 = cos^2 + sin^2 =
1-sin^2 1-cos^2 1
Rules of polynomial functions
1. cannot be divided by a variable 2. the variables (x) can only be 0,1,2,3... (exponents cannot be decimals, sq. rts, neg, or other variables) 3. cannot have infinite number of terms coefficients can be any real or complex (decimals/sq. rt. or i ex/ ix^2+(2+3i)x-17=0 yes)
Mathematical induction proof
1. test for the first possible value (n=1 or even n=0) 2. assume the statement is true for n=k (induction hypothesis) 3. prove the statement is true for n=k+1 with MIP & induction hypothesis
horizontal asymptote 3 rules
1. y=0
2|10 means 2 divided 10 mean
10/2=5 (2*5=10) 2/10 = 0.2
3sq. root of 8 so 8^1/3
2 because to the 3rd is 2*2*2 so just take out one 2
b^2-4ac > 0
2 real solutions
e = number value?
2.71828
Circumference of a circle
2πr
perfectly south east winds on a graph from the x axis will yield what angle?
45 degrees
given determinant a b c d e f g h i = 5 det of g h i a b c d e f = what?
5 when you change one whole row/column with another it multiples the det by (-1) so when you change two rows it will be (-1)(-1)(5)=5
Unit Circle: degrees to radians (Q2) 150 135 120
5pi/6 3pi/4 2pi/3
if 2 polynomials of different degrees (lets say one is 7 degree and the other is 2 degree) what is the maximum points the polynomials can intersect?
7 the potential max points of intersection between the two graphs is based on the highest degree. this is because when you go to solve if two polynomials have any intersections, you set them = to each other and the highest degree once everything is on one side and equals 0 is the highest degree of the polynomial given.
Unit Circle: degrees to radians (Q3) 210 225 240
7pi/6 5pi/4 4pi/3
Multiplicative Inverse of a matrix
A*A^-1=I I= idenity matrix write out matrix A and dot product with same formate matric with variables w, x, y, z and set equal to identity matrix. then solve the system of equations to get w, x, y, z
Compound Interest Formula
A=P(1+r/n)^nt or A=P(1+r)^n A is future amount P initial amount R Interest rate N number of times compounded in one "t" T time 1 is + or - depending on if value is decreasing or increasing
compound continuously formula
A=Pe^rt
Matrix Inverse (2x2)
A^-1 (A) = Identity matrix ONLY FOR 2X2 of A= |a b| |c d| A^-1 = 1/detA|d -b -c a| in the inverse matrix part, switch the top left and bottom right & make the other two variables negative from the original matrix
Absolute value graphs: |y| < -|x|+3 whats the other equation?
Change sign and ALL (+) or (-) to opposite so: |y| > |x|-3
Homogenous System of Linear Equations and Trivial vs Non-trivial solutions think of Ax=B
Homogenous means, each variable must have the same solution for the constant term, this is usually when they all= 0 so Ax=0 x+y+z=0 -2x+4y+z=0 -x+2y-3z=0 Trivial means the solution works has little importance so when the determinate = 0, whereas nontrivial is when the determinate does *not* equal 0
Transitive Property
If a=b and b=c, then a=c why (-x)(-y)=xy
Properties of Determinants
Let A and B be square n x n matrices. a. If a multiple of one row of A is added to another row to produce a matrix B, then det B = det A b. if two rows of A are interchanged to produce B, the det B = -det A c. if one row of A is multiplied by k to produce B, then det B = k(det A) d. det A^T = det A e. det AB = (det A)(det B) f. If A is triangular, the set A is the product of the entries on the main diagonal of A WARNING: in general, det(A+B)≠det A + det B
Synthetic divisi
Make sure to switch the sign of the number you are diving the other numbers with (in the factor) Ex/ x^2+9x+20 / (x+5) —> pull all numbers out and use -5 to do work with
a/b * b/a is what property
Multiplicative inverse
given determinant a b c d e f g h i = 5 det of: 3a 3b 3c 3d 3e 3f 3g 3h 3i still = 5?
NO this now equals 3*3*3=27 27*(5)=135 each time a row/column gets multiplied by k that is one number times the original det
Do irrational numbers satisfy the Identity Property of Addition?
No, because 0 is not an irrational #
exponential growth equation... what is t when a population is first counted??
P(t) = P(initial)e^(rt) When P(t): Population at time P(i): Initial population e: r: Rate t: Time....t=0 when population first counted
Y= sq. Rt. x vs y= sq. Rt. 2x what happens to graph?
The 2 inside the sq. Rt. Give a horizontal stretch, diving all the x's by 2!
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side. If finding the third side, it has to be less than the sum of the two smaller sides & must be greater than the difference of the two smaller sides
Log to the base 0
Undefined
d(a x b)=
a x (db)
Properties of Determinants A= 1 2 3 -2 -3 2 1 2 1
a. when you switch two rows or columns completely and keep the third the same, determinant= -detA b. if one or all rows/columns multiplied by a number (k) the new determinant = 2detA c. when you perform elementary row operations, this will NOT change the determinant (ex/ R3=R1+R3)
solving system of equations what is an empty set?
an empty set is when you solve systems of equations and get 0x+0y+0z=7 0 does not = 7 so this is no solution or an empty set either two planes or lines that are parallel
cos(2o)= cos(2o)= cos(2o)=
cos^2 - sin^2 1-sin^2 - sin^2 = 1-2sin^2 cos^2 - (1-cos^2) = cos^2 -1 + cos^2 = 2cos^2-1 *cos stays positive and sin stays negative
Finding the angle between two vectors (cos)
cosx = (p * q) / (||*p*||||*q*||) where ||*p*|| = (sq root of a1^2 + a2^2 + a3^2...an^2)
Completing the Square Method
divide b by 2 and square it to find c, and then find the root make sure to add to both sides
when a problem is asking you to maximize, what should you think about?
either linear programming or a polabrola
odd function
f(-x)=-f(x) symmetric about origin
even function
f(-x)=f(x) y-axis symmetry
If 𝐴𝑋=B represents a homogeneous linear system of equations in three-dimensional space and det(𝐴)=0, then the solution set to this system is: a. plane, line or empty b. point or line c. plane or line d. point, line, empty
homogeneous means all variables equal 0 so AX=0 and B=0 if det(A)=0 then there are an infinite number of solutions if there are an infinate number of solutions it would have to equal a line or plane.
domain and range of a function and it's inverse is flipped
if F(x) D:(0,5) & R:(5,10) then F^1(x) D:(5,10) & R:(0,5) So the inverse F^-1(x) domain and range can get restricted due to F(x)'s domain and range (be aware with parabolas)
Domain of a radical function
if in the numerator of a function set what is inside greater than or equal to zero if in denominator cannot be set equal, just greater!!
F(x) is a fourth degree polynomial with real coefficients such that: f(x)/(x-3) = q(x)+(8/x-3), what must be true about f(x)
know 3 is not a zero because when it was divided by the polynomial there was a remained of 8. we can solve for f(x) and get f(x)=(x-3)*q(x)+8 let x=3 and we get 8. so the graph is shifted 8 units up, and therefor, (3,8) is on the graph!
Log rules addition & subtraction
log addition: logbX + logbY = logb(X*Y) log subtraction: logbX - logbY = logb(X/Y)
Interest Formula
principal x rate x time
Binomial Theorem how to find coefficients
n!/k!(n-k!)
b^2-4ac < 0
no real roots
in order for a function to have an inverse it must be what kind of function?
one to one
cos^-1(0)
pi/2 or 90 degrees
60 degrees to radians in all quads
pi/3 2pi/3 4pi/3 5pi/3
45 degrees to radians in all quads
pi/4 3pi/4 5pi/4 7pi/4
30 degrees to radians in all quads
pi/6 5pi/6 7pi/6 11pi/6
Unit Circle: degrees to radians (Q1) 30 45 60
pi/6 pi/4 pi/3
Log rules logb(M^p)=
plogbM
finding magnitude of a vector
take the direction <x,y> magnitude = sq. rt. x^2+y^2
two lines that are perpendicular have what kind of slopes?
the slopes are negative reciprocals of each other if one like is -2 the other has to be 1/2
trivial solution and linear independent or dependent?
the solution x=0 of a homogeneous equation Ax=0 means that vector is linear independent
finding how many prime numbers go into an equation ex/ Find the number of primes in the form | n^2-2n-3 | where n is an integer?
think about prime numbers, what does that mean? For a number to be a prime, its only factors can be 1 and itself. Therefore after being factored one of the factors has to = 1, not 0 like usual. That's why they set each equal to 1 separately. so factors are | n-3 | & | n+1 | set both equal to one and see how many solutions there are. | n-3 |= 1 n can be 4, -4, 2, -2 | n+1 |= 1 so n can be 0 use these new n number to plug into original equation to find what prime numbers, -4 wont gives 21 (not a prime) and the rest are 3 or 5 so 3 and/or 5 is the answer.
how to show a set of vectors are linearly independent or dependent? what does this mean? DO THIS PROOF
we need to find the set of solutions to the the vectors (name all set k1 k1 k1, second vector set k2 k2 k2, third set, k3 k3 k3) and add them to create a system of equations that all equal 0. (k1+k2+k3=0 for all places) then solve for the k's and if = a unique solution then the vectors are linearly independent.
horizontal asymptote: slant
when n>m then need to synthetic division to find the asymptote (which will be a line) can only do synthetic division when the coefficient is 1 of the divisor
given: two vectors (u and v) want to find a vector orthogonal to both vectors or also stated as all nonzero vectors perpendicular to both vectors, how to find?
when two vectors are perpendicular/orthogonal this means we are looking for what vector * u and separately * v will = 0. to find this use the cross product of u and v. can check by doing the dot product of the new vector with u then v to make sure the sum is 0.
a^logaX =
x (inverses)
Whats the conjugate of x+iy?
x-iy The 'x' stays the same and the 'y' changes signs
linear programming word problem states, "the number of x's cannot exceed the numbers of y's by more than #" write an equation
x-y<=#
point slope form
y-y1=m(x-x1)
Finding the angle between two vectors (sin)
|a x b| = |a||b| sin
Area of a circle
πr²