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urp:algebra [2021-11-08] nerf_herder |
urp:algebra [2021-12-19] nerf_herder [Functions] |
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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 | ||
+ | |||
+ | **PEMDAS/BODMAS ** - order of operations: Parentheses/Brackets, Exponents/Order, Multiply-Divide, Add-Subtract | ||
==== System of equations==== | ==== System of equations==== | ||
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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 | + | |
- | + | ||
- | 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 ==== | ||
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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 function - only possible if no two values of x produce the same result | ||
- | graphing an inverse: reflection of the graph about the line y=x | ||
- | **horizontal and vertical asymptotes:** | + | **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 | ||
+ | |||
+ | **asymptotes** | ||
y = (quadratic1 of x) / (quadratic2 of x) | y = (quadratic1 of x) / (quadratic2 of x) | ||
* vertical asymptotes (VA) are when denominator goes to zero | * vertical asymptotes (VA) are when denominator goes to zero | ||
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* if n < k : HA = 0 | * if n < k : HA = 0 | ||
* if n = k : HA = a/b | * if n = k : HA = a/b | ||
+ | * oblique (diagonal) - approaches the line y = mx+b | ||
+ | * look halfway down the page here: https://www.mathsisfun.com/algebra/rational-expression.html | ||
+ | * if power of x in numerator > power of x in denominator: asymptote = 0 | ||
+ | * if power of x in numerator = power of x in denominator: asymptote = horizontal, not zero (ratio of largest coefficients) | ||
+ | * if power of x in numerator = 1 less than power of x in denominator: asymptote = oblique (definition of line, from polynomial long division) | ||
+ | * if power of x in numerator > 1 less than power of x in denominator: asymptote = none | ||
====Interest, half life, amortization==== | ====Interest, half life, amortization==== |