vault backup: 2024-08-28 10:39:34

education/math/MATH1050/Exponents/Exponents.md

@@ -0,0 +1,19 @@
+The standard form of an exponential function looks something like this:
+$$ a(b)^{cx-h}+k $$
+- $a$: Vertical stretch/compression
+- $c$ Horizontal stretch/compression
+- $h$: Horizontal translation left or right
+- $k$: Vertical translation up or down
+
+Without stretch:
+$$ a^{x-b}+c $$
+
+A negative exponent is the equivalent of `1/x`, EG
+$$ x^{-2} = \frac{1}{x^2} $$
+An equation in the base form $2^x$ will have an asymptote of $y = 0$. 
+
+Putting a negative in front of something like $2^x$ flips it over the *x axis* ($-2^x$).
+Putting a negative in front of the exponent ($2^{-x}$) flips it over the y axis. 
+
+## Solving Exponents
+To solve an equation that's got variables in the exponents, work on setting the bases on each side equal, then you can treat the exponents like standalone. Refer to the notes on [[Logarithms]] for ways to do that. 

education/math/MATH1050/Exponents/Logarithms.md

@@ -0,0 +1,52 @@
+ https://www.youtube.com/watch?v=sULa9Lc4pck
+
+$$log_a(b) $$
+Pronounced log *base* a, this function is used to figure out what exponent you need to raise $a$ to to get $b$.
+
+$log_ab = c$ can be rewritten as $a^c = b$.
+
+
+$$ 5^{log_5^{(x+2)}}=x+2 $$
+By default, $log$ refers to $log_{10}$. $ln$ is shorthand for $log_e$. 
+
+$$ \sqrt{x} = x^{1/2} $$
+To get the reciprocal of a value, change the sign of the exponent.
+$$ x^{-1} = \frac{1}{x} $$
+## Domain
+There are 3 places you need to worry about domain. The third is specific to logarithms.
+- You can't divide by 0
+- You can't take the square root of a negative without complex numbers
+- You cannot take the $log$ of a zero, or a negative number.
+	- There's no way to raise a number to an exponent and have it equal zero, or be a negative number.
+	- This can be used to help solve inequalities, because you know *an equation that's wrapped in a logarithm must be $> 0$*.
+
+### Finding the domain of added logarithms
+$$ log(x+2) + log(2x-3) $$
+With the above example, you can find the domain of each function separately, then find the overlap of valid numbers.
+
+
+# Expanding logarithms
+$$ log_b(x*y) = log_b x + log_b y $$
+$$log_b(\frac{x}{y}) = log_b x - log_b y $$
+
+Example Problem:
+$$ log_5 z = 3 $$
+$$ log_5 y = 2 $$$$log_5(yz) = log_5 y + log_5 z $$$$ 2 + 3 = 5 $$
+Exponents can be moved to the front of a logarithm
+$$ log_3 x^5 = 5*log_3 x $$
+Roots are just the inverse, so:
+$$ log_3 \sqrt x = \frac{1}{2}*log_3x $$
+## Change of base
+$$ log_b x = \frac{\log x}{\log b} = \frac{\ln x}{\ln b} $$
+The above are all equivalent because the ratios are the same
+
+
+### The compound interest formula
+$$ A= Pe^{rt} $$
+
+| Value | Description |
+| ---- | ---- |
+| $A$ | Ending amount |
+| $P$ or $A_0$ | Starting amount |
+| $r$ or $k$ | Rate (a %) |
+| $t$ | The amount of times interest is compounded |

education/math/MATH1050/Exponents/Quadratics.md

@@ -0,0 +1,25 @@
+# Notes
+
+To convert an equation in the form of $y = (x^2-3x)$ into a square equivalent, you:
+- Take the second value, $3$, then half it, giving you ($\frac{3}{2}$). 
+- Then to rebalance the equation, you're going to square that value (giving you $\frac{9}{4}$), then add it to the other side, multiplying it by $a$ if necessary, where $a$ is what the parentheses are multiplied by in the form $a(x - h)^2$. This will give you an equation that looks something like: $\frac{9}{4} + y = (x -\frac{3}{2})^2$.
+- Finally, you can rebalance the equation by subtracting $\frac{9}{4}$ from both sides, giving you $y = (x - \frac{3}{2})^2 - \frac{9}{4}$. This equation should be equal to the original.
+
+$$ y = -5x^2 -20x + 13 $$
+Given the above equation, you can factor out a -5, resulting in the equation $-5(x^2+4x) + 13)$. Half of 4 is 2, and because the inside is multiplied by -5, $-5 *4 = -20$, so you add -20 to the other side to equalize the equation, resulting in an equation in the form of $-20 + y = -5(x+2)^2+ 13$. This simplifies down to $y = -5(x+2)^2 + 33$. 
+# Forms
+**Standard form (vertex form)**
+$$ y = a(x - h)^2 + k $$
+To convert to standard form given a vertex of a quadratic equation and a point that falls along that line, plug values in for everything and solve for $a$. 
+
+**Quadratic form**
+$$ y = a^2 + bx + c $$
+# End Behavior of functions
+If the largest exponent of a function is **even**, both sides of a function will point the same way.
+Given there's no negative coefficient, as $x$ goes towards infinity, $f(x)$ will go towards infinity. As $f(x)$ goes towards $-\infty$, $f(x)$ will still go to infinity. A negative coefficient will flip this.
+$$ x \rightarrow \infty, \space f(x) \rightarrow \infty $$
+
+$$ x \rightarrow -\infty, \space f(x) \rightarrow -\infty $$
+If the largest exponent of a function is **odd**, each side of the function will point towards a different direction. Given there's no negative coefficient, the left side of the graph will point down, and the right side will point up. A negative coefficient will flip this.
+
+The *least degree* of a polynomial is the number of turning points + 1.