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urp:algebra
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urp:algebra [2021-11-03] nerf_herder [Logs] |
urp:algebra [2021-11-08] nerf_herder |
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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]] | ||
| - | **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 50: | Line 52: | ||
| (nᵃ)ᵇ = nᵃᵇ | (nᵃ)ᵇ = nᵃᵇ | ||
| n⁻ᵃ = 1/nᵃ | n⁻ᵃ = 1/nᵃ | ||
| + | |||
| + | (Note: The square root sign, √, refers only to the //positive// root. Use ± to include both roots.) | ||
| Factorial: | Factorial: | ||
| Line 59: | Line 63: | ||
| ==== Logs ==== | ==== Logs ==== | ||
| - | inverse of a power | + | Log is the inverse of a power |
| x = bᵉ, e = logᵦ(x) (b = base, 10 by default) | x = bᵉ, e = logᵦ(x) (b = base, 10 by default) | ||
| eg. 2³ = 8, log₂(8) = 3 | eg. 2³ = 8, log₂(8) = 3 | ||
| Line 77: | Line 81: | ||
| ==== 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 | ||
| + | |||
| + | **horizontal and vertical 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 : no HA |
| - | if n > m : no HA | + | * if n < k : HA = 0 |
| - | if n < m : HA = 0 | + | * if n = k : HA = a/b |
| - | if n = m : HA = a/b | + | |
| ====Interest, half life, amortization==== | ====Interest, half life, amortization==== | ||
| Line 113: | Line 129: | ||
| ====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.txt · Last modified: 2022-02-01 by nerf_herder
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