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urp:algebra [2021-11-03] nerf_herder |
urp:algebra [2022-02-01] (current) 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 | + | **PEMDAS/BODMAS ** - order of operations: Parentheses/Brackets, Exponents/Order, Multiply-Divide, Add-Subtract |
| ==== System of equations==== | ==== System of equations==== | ||
| Line 33: | Line 36: | ||
| https://courses.lumenlearning.com/suny-osalgebratrig/chapter/parametric-equations/ | https://courses.lumenlearning.com/suny-osalgebratrig/chapter/parametric-equations/ | ||
| - | ==== Arithmetic Sequences ==== | + | ==== Arithmetic/Geometric Sequences ==== |
| + | |||
| + | **Arithmetic sequence** has a constant difference between the terms, such as 1, 3, 5, 7, 9... | ||
| 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 | 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 | ||
| Line 43: | Line 48: | ||
| S = n/2 (2a + (n − 1) d) | S = n/2 (2a + (n − 1) d) | ||
| - | ==== Powers & Radicals ==== | + | **Geometric sequence** terms are found by multiplying the previous term by a constant, such as 2, 4, 8, 16 ... |
| - | Combining powers | + | a(n) = arⁿ⁻¹ |
| - | nᵃnᵇ = nᵃ⁺ᵇ | + | |
| - | nᵃ/nᵇ = nᵃ⁻ᵇ | + | |
| - | (nᵃ)ᵇ = nᵃᵇ | + | |
| - | n⁻ᵃ = 1/nᵃ | + | |
| - | Factorial: | + | Other sequences exist: |
| + | * squares: a(n) = n², cubes, etc. | ||
| + | * triangular numbers: a(n) = n(n+1)/2 (number of dots in a triangle of n rows) | ||
| + | * fibonacci sequence: a(n) = a(n-1) + a(n-2) | ||
| + | |||
| + | Convergence of a power sequence: http://math.bu.edu/people/prakashb/Teaching/32LS10/Lectures/11-2.pdf | ||
| + | ====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) = 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. | ||
| - | eᵏ = 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) |
| + | 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 |
| - | horizontal asymptotes (HA) is when x goes to infinity, | + | * horizontal asymptotes (HA) is when x goes to infinity, look at highest order of x in numerator and denominator: |
| - | look at highest order of x in numerator and denominator: | + | * y = axⁿ / bxᵏ |
| - | y = ax^n / bx^m | + | * if n < k : HA = 0 |
| - | if n > m : no HA | + | * if n = k : HA = a/b |
| - | if n < m : HA = 0 | + | * if n > k : no HA |
| - | if n = m : HA = a/b | + | * if n = k+1 : oblique (diagonal) asymptote - approaches the line y = mx+b (from polynomial long division) |
| + | * if n > k+1 : no asymptote | ||
| + | |||
| + | look halfway down the page here: https://www.mathsisfun.com/algebra/rational-expression.html | ||
| ====Interest, half life, amortization==== | ====Interest, half life, amortization==== | ||
| Line 113: | Line 116: | ||
| ====misc==== | ====misc==== | ||
| abs. value of imaginary number | abs. value of imaginary number | ||
| - | |a + bi| = sqrt(a^2 + b^2) | + | |a + bi| = √(a^2 + b^2) |
| __rational numbers__ can be expressed as a fraction of two integers. The decimal expansion either terminates or repeats. | __rational numbers__ can be expressed as a fraction of two integers. The decimal expansion either terminates or repeats. | ||
urp/algebra.1635904537.txt.gz · Last modified: 2021-11-03 by nerf_herder
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