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urp:algebra [2021-11-01] nerf_herder [Powers & Radicals] |
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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==== | ||
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Combining powers | Combining powers | ||
- | (nᵃ)(nᵇ) = nᵃ⁺ᵇ | + | nᵃnᵇ = nᵃ⁺ᵇ |
- | (nᵃ)/(nᵇ) = nᵃ⁻ᵇ | + | nᵃ/nᵇ = nᵃ⁻ᵇ |
(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: | ||
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==== Logs ==== | ==== Logs ==== | ||
- | inverse of a power | + | Log is the inverse of a power |
- | x = b^y, y = logb(x) (b = base, 10 by default) | + | x = bᵉ, e = logᵦ(x) (b = base, 10 by default) |
- | eg. 2^3 = 8, log2(8) = 3 | + | eg. 2³ = 8, log₂(8) = 3 |
log(1) = 0 (for any base), log(x) is undefined for x =< 0 | log(1) = 0 (for any base), log(x) is undefined for x =< 0 | ||
- | logb(x) = logk(x) / logk(b) | + | logₐ(x) = logᵣ(x) / logᵣ(a) |
- | = log(x) / log(b) for k=10 | + | = log(x) / log(a) for r=10 |
- | = ln(x) / ln(b) for k=e (e = Euler's number, 2.718..) | + | = ln(x) / ln(a) for r=e (e = Euler's number, 2.718..) |
ln(e) = 1, log(10) = 1 | ln(e) = 1, log(10) = 1 | ||
- | ln(x)=logex. | + | ln(x) = logₑx. |
- | e^k = c, and k = ln(c) => e^ln(c) = c | + | eᵏ = c, and k = ln(c) => e^ln(c) = c |
- | a^loga(x) = x (power and log are inverses, cancel each other out) | + | a^logₐ(x) = x (power and log are inverses, cancel each other out) |
- | loga(a^x) = x (same reason) | + | logₐ(aᵏ) = k (same reason) |
product rule: log(ab) = log(a) + log(b) | product rule: log(ab) = log(a) + log(b) | ||
quotient rule: log(a/b) = log(a) - log(b) | quotient rule: log(a/b) = log(a) - log(b) | ||
- | power rule: log(a^b) = b*log(a) | + | power rule: log(aᵇ) = 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 | + | **horizontal and vertical 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 | ||
+ | ====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) | ||
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(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 ==== |