SAT Math 2 Mega Quizlet

Vertex of Quadratic

(-b/2a, f(-b/2a))

Order of a Matrix

(MxN) where m is the number of rows and n is the number of columns

a^3+b^3

(a+b)(a^2-ab+b^2)

a^3-b^3

(a-b)(a^2+ab+b^2)

Midpoint formula on a coordinate plane

(x₁+x₂)/2, (y₁+y₂)/2

Rational Root Theorem

+/- p/q Where P is the factors of the constant Q is the factors of the leading coefficient

If E is an event, then

0<= P(E) <= 1 P(not E)= 1-P(E), If P(E)=0 the event is known as an impossible event, If P(E)=1 the event is known as a certain event.

A point , (r, Θ) can be represented in different ways...

1) (r, Θ+2pi n) 2) (-r, Θ+ (2n-1)pi) n is an integer

Distance formula on a coordinate plane

2d---d = √[( x₂ - x₁)² + (y₂ - y₁)²] 3d---d =√[( x₂ - x₁)² + (y₂ - y₁)² +(z₂-z₁)²] 3d shapes √l²+w²+h²

P (A or B)

= P(A) + P(B) - P(A and B)

Types of Symmetry

A graph is.. Symmetrical to the x axis if replacing x by -x preserves the equation Symmetrical to the y axis if replacing y by -y preserves the equation Symmetrical to the origin if replacing x and u by -x and -y preserves the equation

Fundamental Theorem of Algebra

A polynomial will have as many roots as its degree

Defn Ellipse

An Ellipse is the set of all points in a plane, the sum of whose distance from two distinct fixed points (foci) is constant.

A Matrix times its inverse yields...

An identity matrix. An identity matrix times a matrix, say matrix A, yields Matrix A again.

Exponential Properties

And Xⁿ × Yⁿ = (XY)ⁿ

Identity Matrix

Any Matrix times its respective Identity Matrix will = the same Matrix

General Equation of a line

Ax+By=C

Functions with Inverses must...

Be one to one( pass vertical and horizontal Line tests.

If the power of a polynomial is even then

Both ends of the graph leave in the same directions

Even Multiplicity

Bounce on X axis

Multiplying Matrices

By a constant: Mult each term in the matrix by that constant(example, (3A- 3 times Matrix A) By another Matrix: See example image You may only multiple by another matrix if this is true: (MxN) X(NxP) =(MxP)* *Columns must equal rows*

Coefficient Matrix vs Augmented

Coefficient Matrix: Derived from the coefficients of a system of equations... no constants, no variables. Augmented Matrix:Derived from a system of equations (Coefficients and Constants)

Consistent vs Inconsistent Systems

Consistent Systems: Exactly one solution( intersection) or Infinitely many solutions ( Coinciding/same lines) Inconsistent System: No Solution(Parallel Lines

Absolute Value Function

Defined as X if x>= 0 or -X of X<0 General form Y=alx-hl +k with vertex (h,k)

Determinants and Singularity

Determinant of a square 2x2 Matrix= ad-bc The Determinants of other Matrix sizes most be found using a calculator. A Square matrix is called singular if the determinant is zero and nonsingular if its determinant is not zero. Only Nonsingular matrices have inverses. The inverse of a nonsingular,A, is defined as a square matrix A⁻¹ of the same size sich that AA⁻¹=A⁻¹A=I

Function Rule

Every X must have exactly one Y value An X value cannot have two Y Values*

Determinant Matrix

For a 2x2 see image For anything greater you must use cofactors and minors(see chapter7.7 notes).

General and Standard Forms of Circles

General Form: AX ² +BY ²+Cx+Dy=E=0 A, B ≠0 A=B Standard Form: (x-h)²+(y-k)²=r² Center (h,k) Radius:r

General and Standard forms for Parabolas

General form: AX²+BY²+Cx+Dy+E=0 A or B=0, but not both Standard form (x-h)²=4p(y-k)->opens up or down OR (y-k)²=4p(x-h) (h,k) is the vertex ** the focus lies on the axis of symmetry p units from the vertex. D

General and Standard forms of Hyperbolas

General form: AX²+BY²+Cx+Dy+E=0 A>0, B<0 OR A<0,B>0 Standard Form: see image If x² is positive there is a horizontal transverse axis(vertical conjugate axis), If y² is positive then there is a vertical Transverse axis(horizontal conjugate axis). The Transverse axis encompasses Foci, Vertices and the center a² is always under the positive variable center for both is (h,k) Vertices are a units from center Foci are c units from center *****c²=a²+b²***** Conjugate axis has a length of 2b, Transverse axis has a length of 2a Asymptotes: (y-k)= ±b/a(x-h)

General and Standard forms of Ellipses

General form: AX²+BY²+Cx+Dy+E=0 A≠B A>0, B>0 Or... A<0 , B<0 Standard form: See image a² is always the larger value, if its under (x-h)² => Horizontal Major Axis, if its under (y-k)² => Vertical Major Axis

Inverse Matrix(B)

If B is the inverse matrix Matrix A times inverse Matrix B=The Identity Matrix. *Not all square matrices have inverses *a Non square matrix will never have an inverse.

When is Matrix multiplication possible?

If Matrix A is r1 by c1, Matrix B is r2 by c2 And c1=r2 --The columns of the 1st matrix= rows of the 2nd then AB is defined and has size r1 by c2 Multiply row by column

Odd Function

If f(-x)=-f(x) for all x in its domain, the graph is symmetric about the origin and is an odd function

Even Function

If f(-x)=f(x) for all x in its domain, the graph is symmetric about the y-axis and is an even function.

Geometric Sequences and Series Equation

Nth Term: An= a₁^r (n-1) Sum of a finite series: Sn= A1(rⁿ-1)/ (r-1) Sum of an infinite series (abs(r) must be <1 then Sn= A1/(1-r)

A<0

Opens Downard

A>0

Opens Upward

P(A | B)

P (A n B)/P(B)

If the P(B) occurring is not affected by P(A) the events a re said to be independent and..

P (A n B)= P(A) x P(B)

P(A and B)

P(A) × P(B)

mutually exclusive events

P(A∩B)=0, meaning the events have no common outcomes.

Order of Processes

Reflect (Negation), Changing the Scale (Multiplication), Translation (Add/Sub)

Order of Transformations

Reflect, Change the scale, then translate.

Defn Parabola

Set of all points in a plane that are equidistant from a fixed line(directrix) and a fixed point (focus). The midpoint btwn the focus and the directrix is called the vertex. The line passing through the vertex and focus is the axis of symmetry.

Types of Square Matrices

Singular: Determinant=0 Nonsingular:Determinant doesn't =0 **Only nonsingular matrices have inverses

Arithmetic Sequences and Series Equations

Sum of a Finite Series :Sₙ=n/2(a₁+an) Nth Term: an= a1 +(n-1)d

**Sum and Product of Roots

Sum: -b/a Product: c/a

Defn Circle

The Set of all points equidistant from a given point-the center-

If the power of a polynomial is odd then

The ends of the graph leave in opposite directions

Norm of a vector

The magnitude of the vector√a²+b² Can be used to find unit vector by dividing the vector coordinates by iy.

Gauss Jordan

The process of making a matrix Reduced Row Echelon Form

Defn Hyperbolas

The set of all points in a plane, the difference of whose distance from two fixed points (foci) is a positive constant

Sample Space (Probability)

The set of all possible outcomes of an experiment.

slopes of vertical lines are _____

Undefined

Rational asymptotes vs holes

Vertical Asymptotes make only the denominator 0 (x=...) Horizontal Asymptotes follow rules If the degree of Numerator= degree of Denominator then Horizontal asymptote is the ratio between the coefficient of the numerator and denominator. (Y=...) If the degree of numerator is less than that of the denominator then the horizontal asymptote is zero If the degree of the numerator is greater than that of the denominator then the function approaches +- ∞, No HA but there are slant asymptotes. HOLES Are discontinuities that occur at a value that make both the numerator and denominator zero.

Parts of an Ellipse

Vertices, foci, and center are all on the same line(Major Axis) Major axis has a length of 2a( a is the distance from the center to either vertex). Minor axis connects center to co-vertices, has a length of 2b(b is the distance from Center to co-vertices) c is the distance from the center to either foci **c²=a²-b²**

Graph Symmetry Rules

X Axis: If replacing x by -x preserves the equation Y Axis: If replacing y by -y preserves the equations Origin: If replacing x and y by -x and -y, respectively, preserves the equation.

On an imaginary coordinate plane

X Axis= Real Part of complex number Y Axis: Imaginary part of complex number

complex number plane

X axis: Real Part, Real Units Y Axis:Imaginary Part, Imaginary Units

Important Equations for Polar

X²+Y²=R² RcosΘ=X RSinΘ=Y

Function Symmetry Rules (For all X)

Y Axis: F(x)=F(-x) X Axis: F(x)= -F(x) Origin: F(x)= -F(-x)

Log Form and Graph

Y= log b(x) All Functions pass through (1,0) IF b>1 the graph increases as x increases and approaches the y axis as an asymptote when x decreases (negative infinity). If 0<b<1, the graph decreases as x increases and approaches the y axis (infinity) as x approaches zero. The curvature of the graphs increase as the value becomes closer to zero.

Adding/ Subtracting Matrices

You may only add/subtract matrices if the matrices have the same order as one another

slopes of horizontal lines are _____

Zero

Modulus of A+Bi

a^2+b^2= Modulus^2

Law of Cosines

a²=b²+c²-2bcCosA

Odd Multiplicity

crosses the x-axis

Odd Functions

f(-x)=-f(x) origin symmetry

Even Functions

f(-x)=f(x) y-axis symmetry

vertex form of a quadratic function

f(x) = a(x-h)^2 + k

Odd Function

graph is symmetrical with respect to the origin; f(-x)=-f(x)

Even Function

graph is symmetrical with respect to the y-axis; f(x) = f(-x)

Absolute Value Transformations

if a given graph of f(x) has negative y values the absolute value will undo all the negative values flipping them over the x axis and making them positive. If f(x) <= 0 l f(x)l =-f(x) Transformations done on x values will have the same y values.

Ti Nspire greatest integer function abbrv

int(x) The greatest integer that is less than or equal to x

Triangle Cases

pgs 71-74

Polar

rcostheta=x rsintheta=y

Even/Odd Identities

sin(-θ) = -sinθ csc(-θ) = -cscθ cos(-θ) = cosθ sec(-θ) = secθ tan(-θ) = -tanθ cot(-θ) = -cotθ

Cofunction Identities

sin(π/2-θ) = cosθ cos(π/2-θ) = sinθ tan(π/2-θ) = cotθ cot(π/2-θ) = tanθ sec(π/2-θ) = cscθ csc(π/2-θ) = secθ

Law of Sines

sinA/a=sinB/b=sinC/c

Multiplicity

the number of times a value occurs as a root of the polynomial.

a function must pass the...

vertical line test

axis of symmetry

x=-b/2a

Point slope form of a line

y - y1 = m (x - x1); where (x1, y1) is a point on the line

Exponential Form

y=a^x where a>0 and a cannot equal 1. The inverse of Y=a^x is x=a^y which reads "y is the power of a that makes a" meaning y=logₙX (n=a) If a>1 then the graph increases and approaches the x axis as an asymptote as x decreases. The curvature of the graph becomes greater the greater the value of a. If 0<a<1, the graph decreases as x increases and approaches the x axis asymptote ALL GRAPHS PASS THROUGH (0,1)

slope-intercept form

y=mx+b

Vector dot product

→V dot→U =v1u1+v2u2 If V dot U is zero the vectors are perpindicular

Modulus of a complex number

√a²+b²