TeXes Mathematics 7-12 (235)

sum of an arithmetic series

#terms (1st term + last term)/2

total number of terms in an arithmetic series

((last term - 1st term) /difference btwn terms) + 1

midpoint of (x₁,y₁) and (x₂,y₂)

((x₁+x₂)/2, (y₁+y₂)/2)

vertex of a parabola

(-b/2a, (-b²/4a)+c)

quadratic formula

(-b±√(b²-4ac))/2a

degree measure of an interior angle of a regular n sided polygon

(180(n-2))/n

parabola axis of symmetry

-b/2a

integers (Z)

...-2,-1,0,1,2...

fibonacci sequence

0,1,1,2,3,5,8,13, 21, etc.

whole numbers (W)

0,1,2,3...

x⁰

1

natural numbers (N)

1,2,3...

inductive reasoning

1. Based on observation 2. Does not necessarily lead to a correct conclusion

standard deviation directions

1. Calculate the average of the numbers 2. For each number, subtract the number from the average and square the result 3. Add all the values from #2 4. Divide the value of #3 by (#terms - 1) 5. Take the square root of value of #4

deductive reasoning

1. Follows from some premise 2. Leads to a correct conclusion

how to find the inverse of a function

1. Reverse the x and y in the original function 2. Solve for x

ways to solve a quadratic equation

1. factoring 2. completing the square 3. quadratic formula 4. graphing

secant (sec)

1/cos

cosecant (csc)

1/sin

cotangent (cot)

1/tan

x⁻ⁿ

1/xⁿ

log₁₀ m = n

10ⁿ = m

nth term of an arithmetic series

1st term + difference btwn terms (#terms -1)

using a graph to determine nature of the solution to a quadratic equation

2 real roots: crosses x axis twice 1 real root: tangent to x axis 0 real roots: neither crosses nor is tangent to x axis

using discriminate to determine nature of the solution to a quadratic equation

<0: 2 complex roots =0: 1 real rational root >0 & perfect square: 2 real rational roots >0 & ∧perfect square: 2 real irrational roots

interest (compounded monthly)

A = P (1+r/n)∧rt A = final amount P = beginning amount r = rate n = # times compounded per year t = # years

acronym for remembering where the trig functions are positive

ACTS

complex conjugate root theorem

For a polynomial, P(x), with real coefficients, if P(x) has a complex root, z, then it must also have the complex conjugate of z as a root.

variance directions

Steps 1-4 of Standard Deviation directions

areas vary as the square of heights

a = kh²

interest (compounded continuously)

a = pe∧rt

fundamental theorem of algebra

a polynomial of degree n must have n roots (which may be real or complex and which may not be distinct). OR An nth degree polynomial has n (not necessarily distinct) zeros.

sum of a finite geometric series

a(1-rⁿ)/(1-r) n = #terms a = 1st term r = common factor

distributative property

a(b+c) = ab+ac

inverse

a+(-a) = 0 a×(1/a) = 1

associative property

a+(b+c) = (a+b)+c a(bc) = (ab)c

identity

a+0 = a a×1 = a

commutative property

a+b = b+a ab = ba

sum of an infinite geometric series

a/(1-r) a = 1st term r = ratio

a sin (bx-c) + d

a: amplitude (taller) b: period (faster) c: phase shift (+: left; -:right) d: vertical shift

orientation of an exponential function, f(x) = abⁿ

a>0 & 0<b<1: function/graph is decreasing a>0 & b>1: function/graph is increasing

derivative of velocity

acceleration

2x2 determinant |AB| |CD|

ad - bc

divisible by 2

all even numbers

integral

area under the curve (opposite of derivative) aka anti-derivative

nth term of a geometric series

arⁿ⁻¹ a = 1st term r = common factor n = #terms

standard equation of a line

ax + by = c

discriminant of a polynomial

b²-4ac

irrational numbers

cannot be expressed as a fraction e.g. π, √2, e

how to find gcf

create a prime factorization tree, then multiply each common prime factor together.

how to find least common multiple/denominator

create a prime factorization tree, then multiply the each factor by the maximum number of times it occurs.

degree of a polynomial

degree of the highest term

1:1 function

each value in the domain corresponds to one value in the range and vice versa; passes both the horizontal and vertical line tests; only type of function to have an inverse

divisible by 5

ends in 0 or 5

divisible by 6

even numbers AND sum of digits is divisible by 3 or 6

odd function

f(-x) = -f(x) (symmetric @ origin)

inverse of a function

f(f⁻¹(x)) = x

even function

f(x) = f(-x) (symmetric @ y axis)

amplitude of a function

half the distance between the minimum and maximum values of the function

rational numbers (Q)

integers & fractions

greatest common factor (GCF)

largest number that is a factor of all the given numbers

log₃4

log 4/log 3

log mn

log m + log n

log m/n

log m - log n

formula relating the measure of an angle and the two arcs it intercepts

m<K= (mMajorArc - mMinorArc)/2

slope of perpendicular lines

multiplied together, they equal -1

log mⁿ

n log m

mⁿ = x

n log m = log x

correlation

negative: as one value increases, the other decreases zero: values are random positive: values increase together

divisible by 4

number formed by last 2 digits is divisible by 4

divisible by 8

number formed by last 3 digits is divisible by 8

real numbers

rational & irrational numbers

complex numbers (C)

real & imaginary numbers

induction proof

show true for initial case (i=1), then show true for next case (i+1)

pythagorean identities

sin²+cos² = 1 1+tan² = sec² 1+cot² = csc²

least common multiple/denominator

smallest number that a group of numbers will divide into. Will be either the largest number given OR a multiple of the largest number.

divisible by 3

sum of digits is divisible by 3

divisible by 9

sum of digits is divisible by 9

degree of a term in a polynomial

sum of the exponents of the variables in that term

phase shift of a function

the amount of horizontal displacement of the function from a given reference point (often the origin).

arithmetic sequence

the difference between successive terms is a constant (addition)

geometric sequence

the difference between successive terms is a factor (multiplication)

modulo

the remainder to a division problem

period of a function

the smallest domain containing one complete cycle of the function

0⁰

undefined

volumes vary as the cube of heights

v = kh³

derivative of a distance

velocity

when a rational function has slant asymptotes

when the degree of the polynomial in the numerator is greater than the degree of that of the denominator

orientation of a parabola

x² = up -x² = down y₂ = right -y² = left

(xy)ⁿ

xⁿyⁿ

xⁿxⁿ

xⁿ⁺ⁿ

xⁿ/xⁿ

xⁿ⁻ⁿ

(xⁿ)ⁿ

xⁿⁿ

formula of a parabola

y = ax² + bx + c

slope intercept equation of a line

y = mx+b

point slope equation of a line

y-y₁ = m(x-x₁)

180⁰

π radians

ⁿ√ⁿ√x

ⁿⁿ√x

x½

√x

√xy

√x√y