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urp:algebra
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urp:algebra [2021-11-01] 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==== | ||
| Line 43: | 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==== | |
| - | (n^a)(n^b) = n^(a+b), (n^a)/(n^b) = n^(a-b) | + | n! = n * (n-1)! |
| - | (n^a)^b) = n^(a*b) | + | 0! = 1 |
| - | n^(-a) = 1/(n^a) | + | 7!/(7-3)! = 7!/4! = 7*6*5 * (4!/4!) = 7*6*5 |
| - | + | ||
| - | Factorial: | + | |
| - | n! = n * (n-1)! | + | |
| - | 0! = 1 | + | |
| - | 7!/(7-3)! = 7!/4! = 7*6*5 * (4!/4!) = 7*6*5 | + | |
| - | + | ||
| - | + | ||
| - | ==== Logs ==== | + | |
| - | + | ||
| - | inverse of a power | + | |
| - | x = b^y, y = logb(x) (b = base, 10 by default) | + | |
| - | eg. 2^3 = 8, log2(8) = 3 | + | |
| - | log(1) = 0 (for any base), log(x) is undefined for x =< 0 | + | |
| - | logb(x) = logk(x) / logk(b) | + | |
| - | = log(x) / log(b) for k=10 | + | |
| - | = ln(x) / ln(b) for k=e (e = Euler's number, 2.718..) | + | |
| - | ln(e) = 1, log(10) = 1 | + | |
| - | ln(x)=logex. | + | |
| - | e^k = c, and k = ln(c) => e^ln(c) = c | + | |
| - | + | ||
| - | a^loga(x) = x (power and log are inverses, cancel each other out) | + | |
| - | loga(a^x) = x (same reason) | + | |
| - | product rule: log(ab) = log(a) + log(b) | + | |
| - | quotient rule: log(a/b) = log(a) - log(b) | + | |
| - | power rule: log(a^b) = b*log(a) | + | |
| ==== Functions ==== | ==== 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 | 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) |
| - | y = (quadratic1 of x) / (quadratic2 of x) | + | Example: To find f⁻¹(x) for f(x) = 5x + 3 |
| - | vertical asymptotes (VA) are when denominator goes to zero | + | y = 5x + 3 |
| - | horizontal asymptotes (HA) is when x goes to infinity, | + | y-3 = 5x |
| - | look at highest order of x in numerator and denominator: | + | x = (y-3)/5 |
| - | y = ax^n / bx^m | + | f⁻¹(x) = (x-3)/5 (replace y with x on the last step, |
| - | if n > m : no HA | + | since x is input to the function, and y is output) |
| - | if n < m : HA = 0 | + | |
| - | if n = m : HA = a/b | + | |
| - | ====misc==== | + | Graphing an inverse: reflection of the graph about the line y=x |
| - | abs. value of imaginary number | + | |
| - | |a + bi| = sqrt(a^2 + b^2) | + | |
| - | rational numbers: can be expressed as a fraction of two integers | + | **asymptotes** |
| - | the decimal expansion either terminates or repeats | + | y = (quadratic1 of x) / (quadratic2 of x) |
| - | irrational includes square roots, pi, etc | + | * 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 | ||
| + | * 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==== | ||
| Annual rate, continuous rate of growth: | Annual rate, continuous rate of growth: | ||
| - | Y = a*b^t | + | Y = a*bᵗ |
| a = principle amount, b = annual growth, t = time (years) | a = principle amount, b = annual growth, t = time (years) | ||
| - | y = a*e^(kt) = a*(e^k)^t, and k = continuous rate of growth | + | y = a*eᵏᵗ = a*(eᵏ)ᵗ, and k = continuous rate of growth |
| k = ln(b) | k = ln(b) | ||
| Line 116: | Line 105: | ||
| (ln(2) is close to 7 => rule of 70 for doubling returns??) | (ln(2) is close to 7 => rule of 70 for doubling returns??) | ||
| + | ====misc==== | ||
| + | abs. value of imaginary number | ||
| + | |a + bi| = √(a^2 + b^2) | ||
| + | |||
| + | __rational numbers__ can be expressed as a fraction of two integers. The decimal expansion either terminates or repeats. | ||
| + | |||
| + | __irrational__ includes square roots, pi, etc | ||
| ==== See also ==== | ==== See also ==== | ||
urp/algebra.txt · Last modified: 2022-02-01 by nerf_herder
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