NonLinear Relationships
Table of Contents
This post includes graphs and lots of formulae. If pageloading is slow, that’s to be expected on older computers (and definitely on phones).
Exponential Functions
 Exponential functions take the form $y=a^{x}$. $a$ is a constant,such as 2, 50, π, $\frac{1}{3}$, etc.
 $a$ can also be a negative value:
 Note the green line in both graphs: this is called an asymptote.
 Asymptotes are lines which a graph will approach, but never reach.
 For example, no matter how small the value of $x$, for the graph $y=2^x$, y will never equal zero (Try it for yourself if you want).
 There are three kinds of asymptotes, but only two we care about right now: vertical and horizontal.
 Horizontal asymptotes are parallel to the xaxis, and are represented as $y=b$, where b is some constant
 Vertical asymptotes are parallel to the yaxis, and are represented as $x=b$, where b is some constant.
 So the graph $y=2^x$ has a horizontal asymptote at $y=0$, meaning that the graph will approach zero, but never reach it.
A slightly more technical definition:
An asymptote of a curve is a line, such that the distance between the curve and the line approaches zero as either the x or y coordinates (or both) tend to infinity.
 Exponential growth occurs when $x$ is greater than 1, while exponential decay occurs when $x$ is less than 1.
Quadratic Functions
 Quadratic Functions take the form $y=ax^{2}+bx+c$, where a, b, and c are constants.
 When graphed, quadratic functions are referred to as “parabolas”.

If $a$ is positive, the parabola has a maximum turning point. If $a$ is negative, the parabola has a minimum turning point.

Quadratics have several properties, which you might be asked about:

Vertex/Turning point $(h)$: $x=\frac{b}{2a}$, substitute the value of x into the original equation to get the y value.

Axis of Symmetry: $x=h$ (this should be a vertical line)

Roots/Zeroes: These are just the xintercepts, and can be calculated using the Quadratic Formula:
$\text{When }ax^2+bx+c=0, x=\frac{b\pm\sqrt{b^{2}4ac}}{2a}$
 Note the $\pm$ in the formula: there can be 0, 1, or 2 roots of a quadratic, depending on the value of the bit under the square root.
 If the bit under the square root is negative, you’ll have no roots, meaning the graph never crosses the xaxis.
 If the bit under the square root is 0, you have 1 root, meaning the vertex is on the xaxis.
 If the bit under the square root is positive, you have 2 roots, meaning the graph cuts the xaxis twice.

Hyperbolas
 Hyperbolas (also known as reciprocal functions) are used to express when two values (x and y) are inversely proportional
 Inversely proportional means that as one value increases, the other decreases, and vice versa.
 Hyperbolas take the form $y=\frac{a}{x}$ where a is a constant.
 Hyperbolas have 2 asymptotes: $x=0$ and $y=0$.
 Take a look at the formula, and you’ll see that if x=0, you’d need to divide by 0, which isn’t a thing.
 The formula can be rearranged to $xy=a$, then to $x=\frac{a}{y}$, and we get the same issue.
 This is usually how we end up with asymptotes: when we need to divide by 0.
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