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urp:algebra
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urp:algebra [2021-11-08] nerf_herder [Functions] |
urp:algebra [2021-12-19] nerf_herder [Arithmetic Sequences] |
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| ====Algebra & Pre-Calc==== | ====Algebra & Pre-Calc==== | ||
| + | Related pages: | ||
| * [[Graphing|Graphing circles, ellipses, parabolas, hyperbolas]] | * [[Graphing|Graphing circles, ellipses, parabolas, hyperbolas]] | ||
| * [[poly|Polynomials, quadratic formula and completing the square]] | * [[poly|Polynomials, quadratic formula and completing the square]] | ||
| + | * [[power|Powers, radicals (roots) & logs]] | ||
| **quadratic formula**: x = (-b ±√(b²-4ac)) / 2a | **quadratic formula**: x = (-b ±√(b²-4ac)) / 2a | ||
| - | **PEMBAS/BODMAS ** - order of operations: Parentheses (brackets), exponents (order), multiply/divide, add/subtract | + | **PEMDAS/BODMAS ** - order of operations: Parentheses/Brackets, Exponents/Order, Multiply-Divide, Add-Subtract |
| ==== System of equations==== | ==== System of equations==== | ||
| Line 45: | Line 46: | ||
| S = n/2 (2a + (n − 1) d) | S = n/2 (2a + (n − 1) d) | ||
| - | ==== Powers & Radicals ==== | + | Convergence of a power sequence: http://math.bu.edu/people/prakashb/Teaching/32LS10/Lectures/11-2.pdf |
| - | + | ====Factorial==== | |
| - | 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)! | n! = n * (n-1)! | ||
| 0! = 1 | 0! = 1 | ||
| 7!/(7-3)! = 7!/4! = 7*6*5 * (4!/4!) = 7*6*5 | 7!/(7-3)! = 7!/4! = 7*6*5 * (4!/4!) = 7*6*5 | ||
| - | |||
| - | |||
| - | ==== Logs ==== | ||
| - | |||
| - | 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 ==== | ==== Functions ==== | ||
| Line 89: | Line 61: | ||
| composition of functions: (f’g)(x) = f(g(x)), order of evaluation is important | 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 | + | **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 | Example: To find f⁻¹(x) for f(x) = 5x + 3 | ||
| y = 5x + 3 | y = 5x + 3 | ||
| Line 97: | Line 69: | ||
| since x is input to the function, and y is output) | since x is input to the function, and y is output) | ||
| - | graphing an inverse: reflection of the graph about the line y=x | + | Graphing an inverse: reflection of the graph about the line y=x |
| **horizontal and vertical asymptotes:** | **horizontal and vertical asymptotes:** | ||
urp/algebra.txt · Last modified: 2022-02-01 by nerf_herder
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