====== Gallery of Curves ======

==== Straight Line ====
{{ straight_line.png|Straight Line}}
$$y = ax + b$$
where 
$a$ is the slope, and\\
$b$ is the y-intercept.
~CLEAR~

==== Conic Section ====

== Circle ==
{{cone_circle.png |Conic Section Circle}}
{{ unit_circle.png|Circle}}
$$x^2 + y^2 = r^2$$
Unit Circle: $$x^2 + y^2 = 1$$

$$(x+a)^2 + (y+b)^2 = r^2$$

~CLEAR~

== Ellipse ==
{{cone_ellipse.png |Conic Section Ellipse}}
{{ ellipse.png|Ellipse}}
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

~CLEAR~
== Parabola ==
{{ parabola.png|Parabola}}
$$y = x²$$
A function with a second order nomial x² will have a parabolic shape,\\
concave if positive, convex if negative

Axis of symmetry.
~CLEAR~

== Hyperbola ==
{{ hyperbola.png|Hyperbola}}

$$y = \frac{1}{x}$$

Not to be confused with [[hyperbole]], an exaggeration used as a figure of speech, like "it weighs a ton".

Applications from wikipedia.\\
Path of sundial shadow.\\
Orbit of asteroid.\\
~CLEAR~

==== Sigmoid ====

{{ sigmoid.png|Sigmoid}}

$$ y = \frac{1}{1+e^{x}}$$

y is always between 0 and 1.\\  
s-shaped curve as in "sigmoid".\\  
asymptotes to 0 to the left and 1 to the right.\\  


~CLEAR~
==== Polynomials ====
{{ polynomial_0_degree.png|Polynomial 0 Degree}}
$$y = x² + 2ax + b$$

{{ polynomial_1_degree.png|Polynomial 1 Degree}}
{{ polynomial_2_degree.png|Polynomial 2 Degree}}
{{ polynomial_3_degree.png|Polynomial 3 Degree}}
{{ polynomial_4_degree.png|Polynomial 4 Degree}}
{{ polynomial_x.png|Polynomial x}}
{{ polynomial_xx.png|Polynomial xx}} |
{{ polynomial_xxx.png|Polynomial xxx}}
{{ polynomial_xxxx.png|Polynomial xxxx}}

~CLEAR~

{{cone_ellipse.png |Conic Section Circle}}
{{cone_ellipse.png |Conic Section Circle}}
{{cone_ellipse.png |Conic Section Circle}}
{{cone_ellipse.png |Conic Section Circle}}
{{quadratic_1.png|Quadratic 1}}
{{quadratic_2.png|Quadratic 2}}
{{quadratic_3.png|Quadratic 3}}

~CLEAR~

===== Spirals =====
{{archimedes_spiral.png|Archimedes Spiral}}
{{logarithmic_spiral.png|Logarithmic Spiral}}
{{helix.png|Helix}}

==== References ====
  * Scripts for these graphs are found in http://samantha.voyc.com/doc/script
  * For more, see https://en.wikipedia.org/wiki/Gallery_of_curves.
  * Khan Academy playlist: https://www.youtube.com/playlist?list=PLQPuPO-ipSuhNoqOUXasI1mAzSXubTv3z

==== Notes ====
Movement\\
Growth\\
Acceleration\\

cone \\
cone sliced by planes \\
conic section: circle, ellipse, parabola, hyperbola

equation of a plane\\
https://www.maplesoft.com/support/help/maple/view.aspx?path=MathApps%2FEquationofaPlane3Points

solving linear equations\\
3 planes in one cube\\
https://en.wikipedia.org/wiki/System//of//linear_equations\\
https://en.wikipedia.org/wiki/Linear_algebra\\

parametric equations\\
http://mathworld.wolfram.com/ParametricEquations.html

linear regression, multiple parameters\\
$$y = x₁w₁ + x₂w₂ + x₃w₃ + b$$