Essential Formulas and Knowledge - Algebra 2

List of most important formulas and concepts that I had trouble understanding.

Formula for finding the nth term of an arithmetic sequence

a^n = a^1 + (n-1)d where d is the common difference and a^1 is the first term and n is the number of terms

BOTH Formulas for finding the sum of an arithmetic series.

* **Arithmetic series formula: ((# of terms) * (first term + last term))/2**
* **Alternate formula: (n(2a+(n-1)d))/2**

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Where n is the number of terms, a is the first term, and d is the common difference.

Formula for finding the nth term of a geometric sequence

ar^(n-1)

Formula for finding the sum of a geometric series.

(a((r^n)-1))) / (r-1)

Formula for an infinite geometric series where |r|

a / (1-r)

Sum of cubes factorization

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

diff of cubes factorization

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

Simple interest formula

(1 + nr/100)(k), where r is the simple % interest rate, n is the number of years, and k is the initial investment

Annual compound interest formula

(k)(1 + r/100)^n, where k is the initial investment, r is the % increase, and n is the number of years.

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Formula for compound interest after n years compounded m times a year.

(k)(1 + r/100m)^nm WHERE R is the percent increase, for example 4.5% goes to 4.5, not 1.045

Formula for infinite/continuous compound interest

k \* e^rt

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e = 2.7128

r = yearly rate

t = time in years

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Keep in mind that the yearly rate is always < 1.

Formula for change of base

log(b) (a) = log(c) a / log(c) b

Explain what each of the following things do:

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f(x) = x

g(x) = (-a)(f(-b(x-h))) + k

\-a flips across the x axis and stretches/compresses vertically by factor of a (a>1 = vertically stretched by factor of a, a

How do you derive abs value inequalities from a number line graph (and how do you find abs value equations from solution sets)?

For (G)OR inequalities (when there is empty space, x goes in opp directions):

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| x - (midpt of empty interval) | > (radius of interval)

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For (L)AND inequalities (filled space between the two points):

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| x - (midpt of filled interval) | < (radius of interval)

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The same goes for solution sets: put them on a number line, find the halfway point of the distance between them, and then | x - (midpt of interval) | = radius of interval

How to solve an equation when there’s two expressions in abs value? (ex. |x| + |x+1| = 4)

Get one abs value expression alone on one side, then split the expression: by the end of this example, you; have four expressions with no abs value in them.

Differentiate real, natural, whole, integers, rational numbers, and irrational numbers

natural: positive integers

whole: nonnegative intergers

integers: integers

rational number: any number that terminates or repeats forever (can be turned into a fraction)

irrational number: number that goes on forever, and thus can’t be turned into a fraction

What is your factoring checkboxes?

• difference of squares

• diff/sum of cubes

• square of binomial

• zero sum pair (add something to make it into a perfect square)

• grouping

• UV substitution

what to do when you have two added variables each being powered to the 1/3 in the denominator? (for ex: 14/(5^1/3 + 2^1/3)

use the sum/diff of cubes factorization to rationalize the denominator

how do you factor by grouping when there’s three terms?

find two numbers that multiply to a\*c and add to b, then split those two numbers in the x and factor the new four-term equation.

what should be your process for finding mins and maxes?

ex: box with maximized area.

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find fencing amount (x+y), then find x in terms of y, then find xy (which turns into a quadratic equation)

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IN A QUADRATIC EQUATION IN STANDARD FORM: -b/2a is the x coordinate of the vertex, then plug that back into the equation to find the y coordinate (max) of the equation (which also finds the vertex for you)

what’s the formula for compound interest as it relates to doubling time?

(k)(b)^t/u

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WHERE k is the initial investment, b is the amount it is being multipled by (eg. doubled, halved, or multiplied by 100), t is the time, and u is the amount of time it takes to double or half the initial investment

What are all the fundamental rules of logs?

• log(b) (xy) = log(b) (x) + log(b) (y)

• log (b) (x/y) = log(b) (x) - log(b) (y)

• log(b) (x^y) = y * log(b) (x)

What is the equation for the absolute value of a complex number

| a + bi |

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=

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(a^2 + b^2)^(1/2)

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its the square root of (a^2 + b^2)

What does it mean when there is a vinculum above a complex number

it means take the CONJUGATE (the flipped sign of the “bi” term)

when finding parabola equations:

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what equation to use when given the vertex? the x intercepts? given three different points? when given the axis of symmetry? when given the roots and they are wacky quadratic formula roots?

* Finding parabola equations
* given the vertex, use vertex form
* Given the x intercepts, use intercept form
* Given three points, use standard form.
* axis of symetry: set up equation with vertex form
* wacky roots: use the sum and product (c/a, -b/a)

what is your first step when completing the square? what is your process?

MAKE SURE THERE IS NO COEFFICIENT OF THE QUADRATIC TERM.

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2d= the linear term.

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d^2 = the constant.

How do you find asymptotes in rational expressions? What’s a rule about horizontal asymptotes? How do you know if theres a hole?

VERTICAL asymptote: whatever can’t be in the domain (set the denominator = to 0 and solve)

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HORIZONTAL:

ax^m/bx^n (delete all but the highest degree on each side

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m
m>n: no horizontal asymptote

m = n: y = a/b

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HORIZONTAL ASMPYOTE: Delete all terms except for the highest degree term on the numerator and denomianator; Then simplify til you get your horizontal asymptote.

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If the numerator is a lower degree than the denominator, then the horizontal asymptote is y = 0.

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Ifr the numerator is at a higher degree than the denominator, there is no horizontal asymptote.

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THERES A HOLE IF THE SAME FACTOR IS ON THE NUMERATOR AND THE DENOMINATOR

What causes all shapes of 2^x?

going up to the right on top: 2^x

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going down to the right on bottom: -2^x

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going down to the right on top: 2^-x

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going up to the right on bottom: -2^-x

what is the #1 thing to remember about logarithmic inequalties?

THE DOMAIN!!!! DO THAT BEFORE YOU START SOLVING