Even vs Odd

Even
$$ f(-x) = f(x) $$
$$ \cos{(-\theta)} = \cos{\theta} $$
Odd
$$ f(-x) = -f(x) $$
$$ \sin{(-\theta)} = -\sin{(\theta)} $$
Example of Neither
$$ f(x) = (x - 1)^2 $$
$$ f(x) = x^2 -2x + 1 $$
$$ f(-x) = (-x)^2 -2(-x) + 1 $$
$$ f(-x) = x^2 + 2x + 1 \ne x^2 - 2x + 1$$
$$ \therefore f(-x) \ne f(x)$$

Functions vs Nonfunctions

Vertical Line Test (VLT)
To test for if it is a function
Horizontal Line Test (HLT)
To test for if it's inverse is a function
Example 1
$$ E = f $$
$$ E^{-1} \ne f^{-1} $$
$$ \therefore function $$
Example 2
$$ E = f $$
$$ E^{-1} = f^{-1} $$
$$ \therefore function $$
Example 3
$$ E = f $$
$$ E^{-1} = f^{-1} $$
$$ \therefore function $$
Example 4
$$ E \ne f $$
$$ E^{-1} = f^{-1} $$
$$ \therefore nonfunction $$

Locus

Locus Definition
Group of points that satisfy a certain condition
Key Words Table
Keywords Math Symbol
is =
# More than +
# Less than -
Abscissa x
Ordinate y
Distance $$\perp$$

Forms, Intercepts, Symmetry, Domains

Forms Table
Form Given Name
$$ y = mx + b $$ m, y-intercept Slope-Intercept Form
$$y - y_1 = m(x - x_1) $$ m, point Point-Slope Form
$$ \frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1} $$ 2 points 2 Point Form
$$ \frac{x}{a} + \frac{y}{b} = 1 $$ x-intercept, y-intercept Intercept Form
$$ x = x_1 $$ || to y || to axis
$$ y = y_1 $$ || to x || to axis
$$ Ax + By = C $$ Standard Form

$$ m = -\frac{A}{B} $$

$$ b = \frac{C}{B} $$

$$ a = \frac{C}{A} $$

Intercept
Point or points that passes x or y axis
Example $$ 9x^2 + 16y^2 = 144 $$
x-intercept (y = 0): $$ 9x^2 + 16(0)^2 = 144 $$
$$ x^2 = \frac{144}{9} $$
$$ x = \pm\sqrt{\frac{144}{9}} = \pm\frac{12}{3} = \pm4, (\pm4, 0) $$
y-intercept (x = 0): $$ 9(0)^2 + 16y^2 = 144 $$
$$ y^2 = \frac{144}{16} $$
$$ y = \pm\sqrt{\frac{144}{16}} = \pm\frac{12}{4} = \pm3, (\pm3, 0) $$
Symmetry
Check via the x-axis, y-axis, and origin
Example $$ 9x^2 + 16y^2 = 144 $$
x-axis (y => -y): $$ 9x^2 + 16(-y)^2 = 144 $$
$$ 9x^2 + 16y^2 = 144 \therefore yes$$
y-axis (x => -x): $$ 9(-x)^2 + 16y^2 = 144 $$
$$ 9x^2 + 16y^2 = 144 \therefore yes$$
Origin / (0, 0), (x => -x, y => -y): $$ 9(-x)^2 + 16(-y)^2 = 144 $$
$$ 9x^2 + 16y^2 = 144 $$
Domain
All x values that yield real y values (solve for y)
Example: $$ 4x^2 + y^2 = 25 $$
And $$ x + y = 5 $$
For this example, ignore the equations used in the graph, I had to convert from implicit to explicit equation
Set one variable equal to the other $$ y = 5 - x $$
Substitute in one equation inside the other $$ 4x^2 + (5 - x)^2 = 25 $$
Solve for the other variable: $$ 4x^2 + 25 - 10x + x^2 = 25 $$
$$ 5x^2 - 10x = 0 $$
$$ x^2 - 2x = 0 $$
$$ x(x - 2) = 0 $$
$$ x = 0, 2 $$
Plug in the x values into one of the equations (I picked the easier one): $$ y = 5 - (0) = 5 $$
$$ y = 5 - (2) = 3 $$
$$ \therefore y = 5, 3 $$
Points of intersection $$ \therefore (0, 5), (2, 3) $$
Range
Pretend like you are finding domain, but switch x for y