Modeling scenarios:

- Every known value can be represented with a number
- Every unknown value can be represented with a variable
- An expression combines multiple values (2*boys - girls, 3x^2, etc.)
- An equals sign creates a balance/equivalence between two or more expressions
- When there is a single variable among numbers in an equation, it is possible to find the numeric value of the variable
- When there are multiple unknown variables, a relationship between the variables can be created to see many possible solutions

Classifying 2D equations:

- One variable mono/polynomial; arithmetic growth:
- linear y = 3x
- quadratic y = 3x^2, x = 3y^2 -3x + 2
- higher order polynomial y = x^3 - 3x^2 + 4
- Inverse of one variable mono/polynomial:
- square root function y = sqrt(x - 3)
- nth root function y = 5 root (4-x)
- Two squared variables in conics:
- circle x^2 + y^2 = 4
- ellipse 3x^2 + y^2 = 4
- hyperbola 3x^2 - y^2 = 4
- Geometric growth/decay:
- exponential y = 2^x, y = e^(2x)
- logarithmic y = log(3x), y = ln(x - 2)
- Trig:
- cyclical waves y = sin(x), y = cos(2x)
- cyclical waves divided y = tan(2x), y = 2cot(3x)
- inverse of cyclical waves y = csc(x), y = sec(x)
- Recursive: a0 = 1, a(n) = 3*a(n-1) + 4

Analyzing number types:

- Real numbers: any number that can be placed on the number line
- Rational numbers: using fractions to represent parts of numbers
- Integers: counting with a way to go positive and negative using whole numbers
- Whole numbers: counting with a way to represent nothing (0,1,2,3,4,5...), and natural numbers (no zero: 1,2,3,4,5...) [US definition]
- Irrational numbers: numbers that can’t be captured as a fraction and never end
- Complex numbers: makes 1D numbers become 2D with a real and imaginary component
- Imaginary numbers: any real number multiplied by the square root of -1, a useful construct for dealing with the common problems caused by negative square roots

Analyzing operations:

- Addition:
- moving left/right on a number line
- with fractions, keep common denominator (factor out the denominator and add the numerators)
- with vectors/complex numbers, add each component separately (factor out the x-components or i-components and add leftovers)
- with polynomials/radicals, treat each power of x or each type of radical as a separate component and add each separately (factor out the common sqrt(5) or x^3 and add leftover coefficients)
- subtraction = addition with second value as a negative
- Multiplication:
- finding area on a grid
- cancel pairs of negatives (a non-obvious rule)
- think “repeated addition”
- with fractions, multiply numerators AND denominators separately
- division = multiplication with second value as an inverse
- Exponentials:
- think “repeated multiplication”
- with fractions, raise numerator AND denominator to a power separately as you would when multiplying
- logarithms = undo an exponential equation to isolate the power term

Equation / function graph transforms [ assuming a form similar to (y-k)/b = (x-h)/a ]

- Translate:
- Move right by subtracting directly from x, move left by adding directly to x
- Move up by subtracting directly from y, move down by adding directly to y
- Reflect:
- Reflect over the y=x line by finding the inverse function (switching x and y)
- Reflect over the y-axis by taking the opposite of all x’s
- Reflect over the x-axis by taking the opposite of all y’s
- Reflect over any vertical or horizontal line by doing a translation, then an axis reflection
- Scale/stretch from origin:
- Divide one side of equation to stretch out along that axis
- Rotate:
- On 90 degree rotations, the x and y coordinates change places and one value becomes negative depending on direction.
- For 0-89 degree rotations, it may be easiest to change to polar coordinates x=rcos(t) and y=rsin(t), then add or subtract from t.

Forms of equations:

- Function:
- solve for y, y = mx + b, y = ax^2 + bx + c
- purpose -- make it easy to find an output given an input, easy to create x-y table
- Expose an important point:
- Point-slope form for linear, y - y1 = m(x - x1)
- Vertex form for quadratic, y = a(x - x1)^2 + y1
- Standard form for circles, (x - x1)^2 + (y - y1)^2 = r^2
- Highlight zeros:
- factored form for polynomials, y = a(x - r1)(x - r2)
- purpose -- make it easy to cancel terms in rational expressions, make it easy to find where graph crosses x-axis (useful in well-designed applications)

Using the coordinate plane:

- Goal: make a problem easier to understand or solve by mapping two different quantities (complex numbers, (x,y) coordinates, latitude/longitude, vectors) onto a visual (requires 2D space)
- How: notice differences between a single answer (a coordinate), a relationship (an equation), and a region of acceptable solutions (an inequality or set of inequalities)
- Use for non-equations: map shapes onto a coordinate plane to more easily find distances and angles

Parameterizing scenarios:

- Functions:
- take an input (usually 1 variable called x)
- have restrictions on what the input can be
- create an output (usually 1 variable called y)
- the output can be classified by the bounds it fits within
- accomplishes something specific
- Domain is the set of input restrictions
- Range is the set of output limitations
- End behavior:
- Purpose: to know which term dictates the long-term behavior and in which direction
- In rational expressions, the highest power term dominates towards positive and negative infinity
- In some expressions, there is an asymptotic line that the output approaches

Accumulation and rates of change:

- Physics:
- Acceleration is rate that the velocity changes (wrt time)
- Velocity is the rate that the position changes (wrt time)
- Position is where something is at a given time
- Rate of change is found using a tangent line of a function at a given point/time
- Accumulation over a period of time is found by measuring the area bounded between a curve and the horizontal-axis.
- Starting with one of the 3 physics graphs, rate of change allows you to move towards acceleration and accumulation allows you to move towards position.