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urp:algebra [2021-10-30] nerf_herder |
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* [[poly|Polynomials, quadratic formula and completing the square]] | * [[poly|Polynomials, quadratic formula and completing the square]] | ||
- | **quadratic formula**: x = (-b +- √(b²-4ac)) / 2a | + | **quadratic formula**: x = (-b ±√(b²-4ac)) / 2a |
- | === System of equations=== | + | **PEMDAS/BODMAS ** - order of operations: Parentheses/Brackets, Exponents/Order, Multiply-Divide, Add-Subtract |
+ | |||
+ | ==== System of equations==== | ||
2 equations, for 2 unknowns (need n equations to solve for n unknowns) | 2 equations, for 2 unknowns (need n equations to solve for n unknowns) | ||
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Example: 5x + 2y = 7, 3x - y = 6 | Example: 5x + 2y = 7, 3x - y = 6 | ||
Solve by a) graphing both equations and see where they intersect (can graph multiple equations on desmos) | Solve by a) graphing both equations and see where they intersect (can graph multiple equations on desmos) | ||
- | b) substitution (using one equation set one variable in terms of the other, | + | b) substitution (using one equation set one variable in terms of the other, the substitute into |
- | then substitute into the other equation): | + | the remaining equation): |
- | y = 3x-6 | + | y = 3x-6 from 2nd equation |
- | 5x + 2(3x-6) = 7 | + | 5x + 2(3x-6) = 7 substitute into first |
- | 11x - 12 = 7 | + | 11x - 12 = 7 solve for x, then can find y |
c) addition - put all like terms in columns, | c) addition - put all like terms in columns, | ||
multiply/divide as needed to get one set of terms to cancel | multiply/divide as needed to get one set of terms to cancel | ||
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11x = 19, solve for x, then put in to find y | 11x = 19, solve for x, then put in to find y | ||
- | === Parametric equations === | + | ==== Parametric equations ==== |
Define x and y in terms of t (time). Arrows on graph represent increasing values of t. Allows you to create functions, using two graphs, from things are not functions for both x and y together (circles, ellipses, x = y², etc) | Define x and y in terms of t (time). Arrows on graph represent increasing values of t. Allows you to create functions, using two graphs, from things are not functions for both x and y together (circles, ellipses, x = y², etc) | ||
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https://courses.lumenlearning.com/suny-osalgebratrig/chapter/parametric-equations/ | https://courses.lumenlearning.com/suny-osalgebratrig/chapter/parametric-equations/ | ||
- | === Powers & Radicals === | + | ==== Arithmetic Sequences ==== |
+ | |||
+ | Basic form to find term n: a(n) = a(1) + d(n-1), where d = step size (difference between terms), a(1) is the first term | ||
+ | |||
+ | Sum of an arithmetic sequence: | ||
+ | S = n/2(a + L) | ||
+ | S = sum, n = # of terms, a = value of first term, L = value of last term | ||
+ | If don't know last term, just substitute a(n): | ||
+ | S = n/2 (2a + (n − 1) d) | ||
+ | |||
+ | ==== Powers & Radicals ==== | ||
+ | |||
+ | Combining powers | ||
+ | nᵃnᵇ = nᵃ⁺ᵇ | ||
+ | nᵃ/nᵇ = nᵃ⁻ᵇ | ||
+ | (nᵃ)ᵇ = nᵃᵇ | ||
+ | n⁻ᵃ = 1/nᵃ | ||
+ | |||
+ | (Note: The square root sign, √, refers only to the //positive// root. Use ± to include both roots.) | ||
+ | |||
+ | Factorial: | ||
+ | n! = n * (n-1)! | ||
+ | 0! = 1 | ||
+ | 7!/(7-3)! = 7!/4! = 7*6*5 * (4!/4!) = 7*6*5 | ||
+ | |||
+ | |||
+ | ==== Logs ==== | ||
+ | |||
+ | Log is the inverse of a power | ||
+ | x = bᵉ, e = logᵦ(x) (b = base, 10 by default) | ||
+ | eg. 2³ = 8, log₂(8) = 3 | ||
+ | log(1) = 0 (for any base), log(x) is undefined for x =< 0 | ||
+ | logₐ(x) = logᵣ(x) / logᵣ(a) | ||
+ | = log(x) / log(a) for r=10 | ||
+ | = ln(x) / ln(a) for r=e (e = Euler's number, 2.718..) | ||
+ | ln(e) = 1, log(10) = 1 | ||
+ | ln(x) = logₑx. | ||
+ | eᵏ = c, and k = ln(c) => e^ln(c) = c | ||
+ | |||
+ | a^logₐ(x) = x (power and log are inverses, cancel each other out) | ||
+ | logₐ(aᵏ) = k (same reason) | ||
+ | product rule: log(ab) = log(a) + log(b) | ||
+ | quotient rule: log(a/b) = log(a) - log(b) | ||
+ | power rule: log(aᵇ) = b*log(a) | ||
+ | |||
+ | ==== Functions ==== | ||
+ | Definitions: | ||
+ | * Function: only has one y value for any x value. Discontinuities are okay (breaks in allowed x values) | ||
+ | * One-to-one function: a function with only one x value for any y value. | ||
+ | * Domain: what values of x are described | ||
+ | * Range: resulting values of y coming from the function | ||
+ | |||
+ | composition of functions: (f’g)(x) = f(g(x)), order of evaluation is important | ||
+ | |||
+ | **Inverse** of a function - only possible if no two values of x produce the same result, ie. must be one-to-one (or limit the domain to make it so) | ||
+ | Example: To find f⁻¹(x) for f(x) = 5x + 3 | ||
+ | y = 5x + 3 | ||
+ | y-3 = 5x | ||
+ | x = (y-3)/5 | ||
+ | f⁻¹(x) = (x-3)/5 (replace y with x on the last step, | ||
+ | since x is input to the function, and y is output) | ||
+ | |||
+ | Graphing an inverse: reflection of the graph about the line y=x | ||
+ | |||
+ | **horizontal and vertical asymptotes:** | ||
+ | y = (quadratic1 of x) / (quadratic2 of x) | ||
+ | * vertical asymptotes (VA) are when denominator goes to zero | ||
+ | * horizontal asymptotes (HA) is when x goes to infinity, look at highest order of x in numerator and denominator: | ||
+ | * y = axⁿ / bxᵏ | ||
+ | * if n > k : no HA | ||
+ | * if n < k : HA = 0 | ||
+ | * if n = k : HA = a/b | ||
+ | |||
+ | ====Interest, half life, amortization==== | ||
+ | Annual rate, continuous rate of growth: | ||
+ | Y = a*bᵗ | ||
+ | a = principle amount, b = annual growth, t = time (years) | ||
+ | y = a*eᵏᵗ = a*(eᵏ)ᵗ, and k = continuous rate of growth | ||
+ | k = ln(b) | ||
+ | |||
+ | Half life: | ||
+ | Nt = No(1/2)^(t/t.5) | ||
+ | Nt = amount at time t, No = initial amount, t.5 = half-life time | ||
+ | This can be rearranged to: | ||
+ | t.5 = t/(log0.5(Nt/No)) = t / (log(Nt/No)/log(1/2)) | ||
+ | Also, | ||
+ | Nt = No*e^(-t/tau) | ||
+ | tau = mean lifetime | ||
+ | tau = 1/lambda, lambda = decay constant | ||
+ | t.5 = ln(2)/lambda | ||
+ | (ln(2) is close to 7 => rule of 70 for doubling returns??) | ||
+ | ====misc==== | ||
+ | abs. value of imaginary number | ||
+ | |a + bi| = √(a^2 + b^2) | ||
- | === Logs === | + | __rational numbers__ can be expressed as a fraction of two integers. The decimal expansion either terminates or repeats. |
- | === Functions === | + | __irrational__ includes square roots, pi, etc |
==== See also ==== | ==== See also ==== |