Algebra Notes

Basic types of equations

Definition of equation forms

Factoring with Exponents

Important Equations

Un-simplified vs Simplified equations

Exploring Radical Equations

Types of Math Equations

• One to One (injective) functions
  - A function where each element of the domain maps to a unique element in the codomain: $f(x)=2x+3$
• Many to One functions
  - A function where multiple elements of the domain may map to the same element in the codomain: $f(x)=x^2$
• Onto (Surjective) function
  - A function where every element in the codomain is mapped to by at least one element in the domain: $f(x)=x$
• Into (Non-surjective) Function
  - A function that does not cover the entire codomain; there are elements in the codomain that are not mapped to by any element in the domain: $f(x)=sin(x)$

Methods of Factoring

Factoring out the Greatest Common Factor (GCF)

Looking for something to factor out of a polynomial is typically the first step in factoring.
Factor $-20x^2y+16xy^3$:
 ①   $4xy(-5x) + 4xy(4y^2)$
 ②   $4xy(-5x+4y^2)$ « Factored

Factor by Grouping

Factoring by grouping is a method of factoring that I typically find less process-based.
Factor $12xy - 8x - 3y + 2$:
 ① (Re-arrange):          $12xy - 3y$  |  $-8x + 2$
 ② (Factor each half):  $3y(4x-1)$  |  $-2(4x-1)$
 ③ Factored » $(3y - 2)(4x - 1)$

1. Get equation into the form of $ax^2 +bx + c$

General Factoring

Factor a trinomial with a leading coefficinet of one you find:
 ①:  Two numbers that add to the middle term.
 ②:  Two numbers that multiply to the constant term.

Factoring, the AC method

When factoring a trinomial with a leading coefficient not equal to one (in the form \(ax^{2}+bx+c\), where \(a\ne 1\)), you begin by multiplying the leading coefficient (\(a\)) and the constant coefficient (\(c\)) together. This is the first step of the "AC method" or factoring by grouping.

 ①:  Find two numbers that multiply to the product \(ac\) and add to the middle coefficient \(b\).
 ②:  Rewrite the original trinomial by splitting the middle term (\(bx\)) into two terms, using the two numbers you found in the previous step. This will give you a four-term polynomial.
 ③:  Factor by grouping by finding the greatest common factor (GCF) of the first two terms and the last two terms.
 ④:  Factor out the common binomial that results from the grouping to get your final answer. 

Example ① - Factor the equation: $20x^2+19x+3$, remembering $y=ax^2+bx+c$:

①Find two values, $e$ and $f$, that allow $e*f=a*c$ to be true as well as $e*f=b$.
inspecting further: $e*f=a*c=20*3$ and $e+f=b=19$
Therefore defining the equations: $e*f=60$ and $e+f=19$

②This lets us re-write the equation as: $20x^2+15x+4x+3$
and then: $5x(4x+3)+1(4x+3)$
finally: $(5x+1)(4x+3)$

Example ② - Factor the equation: $2x^2+3x-9$, remembering $y=ax^2+bx+c$:

①Find two values, $e$ and $f$, that allow $e*f=a*c$ to be true as well as $e*f=b$.
Therefore defining the equations: $e*f=2*-9=-18$ and $e+f=3$

②This lets us re-write the equation as: $2x^2+6x-3x-9$
and then: $2x(x+3)-3(x+3)$
finally: $(2x-3)(x+3)$

③Checking using the Quadratic equation: $x = \frac{-b ± \sqrt{b^2 -4ac}}{2a}$
$\frac{-3±\sqrt{(3)^2-4*2*(-9)}{2*2}}
finally: $(2x-3)(x+3)$

Factoring, Quadratic Formula

The Quadratic Formula: $x = \frac{-b ± \sqrt{b^2 -4ac}}{2a}$

①Starting with an equation's standard form: ax²+bx+c=y
②Set the equation (ax²+bx+c=0) equal to zero to satisfy the Zero Product Property.
③Recall the Quadratic Formula: $x = \frac{-b ± \sqrt{b^2 -4ac}}{2a}$
④Solve for the x-intercepts (x values for when y is zero)
or (x - (???))(x - (???))=0 in factor form.

Then use the Quadratic Formula: $x = \frac{-b ± \sqrt{b^2 -4ac}}{2a}$ to solve for zero locations or (x - (???))(x - (???))=0 in factor form.
Note: the quantity $b^2 -4ac$ is called the discriminant (more below).

Properties of Quadratic Equations

Graphing Quadratic Equations

Finding Max/Min or Vertex of Parabolas

Aka: finding the maximum or minimum value of a quadradic equation.
When the equation is in standard form:
 ①   ax² + bx + c = 0.
 ②   The vertex's $x$ value is: $\frac{-b}{2a}$.
 ③   To get the $y$ value, plug in $x$.

When the equation is in Vertex form:
 ①   $y = a(x-h)^2 + k$     An example being: $y = 2(x-1)^2 + 2$
 ②   The vertex is at: $(h,k)$    ->    Example vertex is at: $(1, 3)$

IMPORTANT:
 ①   In example: $y = 2(x+3)^2 +4$
 ②   Re-write as: $y = 2(x-(-3))^2 + 4$
 ③   Following $(h,k)$, vertex is at: $(-3, 4)$

What is the discriminant?

In the Quadratic Formula: $x = \frac{-b ± \sqrt{b^2 -4ac}}{2a}$
The quantity $b^2 -4ac$ is called the discriminant.

The discriminant is frequently denoted by a Δ where Δ = b² - 4ac.

The discriminant reveals the nature of roots:
    Δ > 0 & Square number: The equation has two distinct real and rational solutions/roots.
    Δ > 0 & Non-square number: The equation has two distinct real and irrational solutions/roots.
    Δ = 0: The equation has exactly one real and rational solution (a repeated root).
    Δ < 0: The equation has two complex (non-real) solutions.

Analyzing: 2x²+9x-5

The equation intersects with the X axis two times.
    2x²+9x-5 factored is: $(2x-1)(x+5)$
    Meaning there are roots at $x=-5$ and $x=\frac{1}{2}$.
    Aka these are points: $(-5,0)$ and $(\frac{1}{2},0)$.
    Vertex form: $2(x+\frac{9}{4})^2 - \frac{121}{8}$.
    Equation has a vertex at: $(-\frac{9}{4},-\frac{121}{8})$.

Analyzing: 4x²-20x+25

The equation intersects with the X axis one time. Parabola opens upward, a>0.
    4x²-20x+25 factored is: $(2x-5)^2$
    Meaning there is one root at $x=\frac{5}{2}$.
    Aka these are points: $(\frac{5}{2},0)$.
    The vertex's $x$ value is: $\frac{-b}{2a}=\frac{-(-20)}{(2*4)}=\frac{20}{8}=\frac{5}{2}$.
    The vertex's $y$ is: $y=4*\frac{25}{4}-20*\frac{5}{2}+25=25-50+25=0$.
    Equation has a vertex at: $(\frac{5}{2},0)$.

Analyzing: 3x²+2x+9

The equation does not cross the X axis. Parabola opens upward, a>0.
    Two complex roots: $x=-\frac{1}{3}±\frac{i\sqrt{26}}{3}$
    Vertex form: $3(x+\frac{1}{3})^2+\frac{26}{3}$
    Vertex at $(-\frac{1}{3},\frac{26}{3})$

Function Family Graphs

Family: Polynomials

Sub-Family: Linear Polynomials, y=x

Sub-Family: Quadratic, y=x^2

Sub-Family: Cubic, y=x^3

Family: Absolute Value

Family: Radical

Sub-Family: Square Root, y=sqrt(x)

Sub-Family: Cube Root, y=x^(1/3)

Family: Exponential

A function of the form $f(x)=a^x$ where a is a constant base and x is the variable exponent. The base $a$ must be a positive real number and $a\ne 1$.

Exponential functions typically have a horizontal asymptote but they do not have a vertical asymptote. The basic form of an exponential function, $f(x)=a(b^x)+d$ has a horizontal asymptote at the line $y=d$ which is the value added or subtracted outside the exponential term. The graph of an exponential function approaches this horizontal line but never crosses it.

Sub-family: $y=b^x$ with base $b, b > 1$

Sub-family: $y=b^x$ with base $b, 0 < b < 1$

Family: Logarithmic Equations

The expression $log_b(a)=c$ is a logarithmic equation that defines the relationship between a base $b$, and an argument $a$, and an exponent $c$. This equation is equivalent to the exponential form $b^c=a$. Again:
$b$: Base
$c$: Exponent
$a$: Argument

Sub-Family: $y=\log_{2}(x)$ (base greater than 1)

Sub-Family: $y=\log_{\frac{1}{2}}(x)$ (base less than 1)

Family: Logarithmic Equations

Sub-Family: Reciprocal, $y=\frac{1}{x}$

Sub-Family: Squared Reciprocal, $y=\frac{1}{x^2}$

Additional Graphs

Exponential and Logarithmic functions are inverses of each other (mirror images across the line $y=x$).

Natural Log Equations

Natural log equations ($ln$ and $ln(x)$) work with Euler's number $e$.
$e = (1+\frac{1}{n})^n$ as $n \to \infty$ approximately equals $\approx 2.718...$

The following property is true:
$log_e(x)=ln(x)$

Graphing Factored Polynomials

Poles & Zeros (middle graph behavior)

Multiplicities or the power of the factor provides crossing behavior of a polynomial.
Referencing the equation $f(x)=(x-c)^m$

* If $m$ is odd, the graph of $f$ crosses the x-axis at $x=c$ (ref: $y=x^3$).
* If $m$ is even, the graph of $f$ touches but does not cross the x-axis at $x=c$ (ref: $y=x^2$).
* When a zero has a multiplicity of 1 the graph of $f$ passes through the x-intercept like in $y=x$.

Following an Example:

A quick way to graph a polynomial is to get it into a factored form:
$y=6(2-x)^3(x+7)^4(x+60)^2(x-0)^1$

» The overall shape of the graph will follow $y=\approx6x^10$, it will be an upside down parabola.

Find the x-intercepts: (x,y) and (multiplicity::behavior)
① y=2-x: (2,0) and (multiplicity of 3, crosses)
② y=x+7: (-7,0) and (multiplicity of 4, bounces)
③ y=x+60: (-60,0) and (multiplicity of 2, bounces)
④ y=x: (0,0) and (multiplicity of 1, crosses)
Find the y-intercept (plug 0 in for x): @(0,0)
utilizing Y-intercepts, X-intercepts, and Multiplicities,

Another complicated equation:

End of Graph Behavior: As $x\rightarrow±\infty$, $f(x)\rightarrow?$

Rational Function "Sign Charts"

For equations like:
$y=\frac{2(x-1)}{(x-4)(x+2)^2}$
Draw a line plot indicating the key values of -2, 1, and 4.

Putting equations into Vertex form from Standard form

Standard form: $y=ax^2+bx+c$
Vertex form: $y=a(x-h)^2+k$

Starting with $f(x)=x^2-2x-3$

Calculating Vertices $x^2$

$vertex(\frac{-b}{2a},f(\frac{-b}{2a}))$ given an equation: $y=ax^2+bx+c$ and ($a != 0$).

Exploring the equation: $g(x)=3x^2-18x-96$

Given: $g(x)=3x^2-18x-96$
$a=3$,$b=-18$,$c=-96$ in $y-ax^2+bx+c$ (standard form).
Therefore, x coordinate: -3 ($\frac{-b}{2a}=\frac{-(-18)}{2(3)}=3$)
Plugging in, $g(3)=3*9-54-96=-123$
Making the vertex: (3,-123)

Analyze equation: $p(x)=3x^2+15x-4$

① The paraola opens downward
② $a=3$,$b=15$,$c=-4$ in $y-ax^2+bx+c$ (standard form).
③ The vertex $vertex(\frac{-b}{2a},f(\frac{-b}{2a}))$ is: $(-5/2,-91/4)$ making $h=-5/2$ and $k=-91/4$
④ There are x-intercepts at: $(-\frac{5}{2}+\frac{sqrt(\frac{91}{3})}{2},0)$ and $(-\frac{5}{2}-\frac{sqrt(\frac{91}{3})}{2},0)$ due to solving the vertex form equation: $v(x)=3(x+\frac{5}{2})^2-\frac{91}{4}$
⑤. There is a y-intercept at: $(0,-4)$ by plugging x=0 into $p(x)=3x^2+15x-4$, $p(0)=0+0-4=-4$
⑥. The axis of symmetry is $x=-5/2$
⑦. Function $p(x)$ has a local minimum value of $-\frac{91}{4}$
⑧. The domain is: $(-\infty, \infty)$ and range is: $[-\frac{91}{4}, \infty)$

Analyze $f(x)=2x^2+3$

① The paraola opens downward
② $a=2$,$b=0$,$c=3$ in $y-ax^2+bx+c$ (standard form).
③ The vertex $vertex(\frac{-b}{2a},f(\frac{-b}{2a}))$ is: $(0,3)$ making $h=0$ and $k=3$
④ There are therefore no x-intercepts because $v(x)=a(x-h)^2+k=2(x-0)^2+3$ when set $y=0$ and solve for x results in $x=±sqrt(\frac{-3}{2})$ or $x=±i*sqrt(\frac{3}{2})$.
⑤. There is a y-intercept at: $(0,3)$ by plugging x=0 into $p(x)=2x^2+3$, $p(0)=0+3=3$
We could thereretically graph the equation at this point because we have two points and the function family is $y=x^2$. Pt1: (0,3), (not yet...) Pt2: (1,5)
⑥. The axis of symmetry is $x=0$
⑦. Function $p(x)$ has a local minimum value of 3 at $(0,3)$
⑧. The domain is: $(-\infty, \infty)$ and range is: $[3, \infty)$

Word problems...

The mothly profit for a company that makes decorative picture frames depends on the price per frame. The company determines that the profit is approximated by $f(p)=-80p^2+2720p-19200$ where $p$ is the price per frame and $f(p)$ is the monthly profit based on that price.
(a) Find the price that generates the maximum profit.
(b) Find the maximum profit.
(c) Find the price(s) that would enable the company to break even. If there is more than one price, use the "and" button.
② $a=-80$,$b=2720$,$c=-19200$ in $y-ax^2+bx+c$ (standard form).
③ The vertex $vertex(\frac{-b}{2a},f(\frac{-b}{2a}))$ is: $(17,3920)$ making $h=17$ and $k=3920$
Therefore, best price is: (a)=$17$. Maximum profit: (b)=$3920$
The breakeven price is (solve for x-intercepts) $v(x)=a(x-h)^2+k = -80(x-17)^2+3920$. Solve $v(x)=0$ $x±sqrt(\frac{-3920}{-80})+17$ or 24 and 10

Rational Root Theorem

The Rational Root Theorem states that for a polynomial equation with integer coefficients, any potential rational root (or zero) can be expressed as a fraction $\frac{p}{q}$, where:
 ①:  $p$ is a factor of the constant term (the term without a variable).
 ②:  $q$ is a factor of the leading coefficient (the coefficient of the term with the highest degree).

To find all possible rational roots of a polynomial equation, you would:
 ①:  List all factors of the constant term (p).
 ②:  List all factors of the leading coefficient (q).
 ③:  Form all possible fractions $\frac{p}{q}$ using the factors from steps 1 and 2.
 ④:  Include both positive and negative versions of these fractions.

These fractions represent all potential rational roots of the polynomial. To determine which of these are actual roots, you would substitute them back into the polynomial equation and check if they satisfy the equation (i.e., make the equation equal to zero).

Dictionary

Q&A

LaTeX

$\frac{\infty}{-\infty}$
$\log_{\frac{1}{2}}(x)$
$\sqrt[2]{x}$