Table of Contents
This post includes graphs and lots of formulae. If page-loading is slow, that’s to be expected on older computers (and definitely on phones).
Exponential Functions
- Exponential functions take the form
. is a constant,such as 2, 50, π, , etc.
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
, for the graph , 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 x-axis, and are represented as
, where b is some constant - Vertical asymptotes are parallel to the y-axis, and are represented as
, where b is some constant. - So the graph
has a horizontal asymptote at , meaning that the graph will approach zero, but never reach it.
- Horizontal asymptotes are parallel to the x-axis, and are represented as
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
is greater than 1, while exponential decay occurs when is less than 1.
Quadratic Functions
- Quadratic Functions take the form
, where a, b, and c are constants. - When graphed, quadratic functions are referred to as “parabolas”.
If
is positive, the parabola has a maximum turning point. If is negative, the parabola has a minimum turning point.Quadratics have several properties, which you might be asked about:
Vertex/Turning point
: , substitute the value of x into the original equation to get the y value.Axis of Symmetry:
(this should be a vertical line)Roots/Zeroes: These are just the x-intercepts, and can be calculated using the Quadratic Formula:
- Note the
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 x-axis.
- If the bit under the square root is 0, you have 1 root, meaning the vertex is on the x-axis.
- If the bit under the square root is positive, you have 2 roots, meaning the graph cuts the x-axis twice.
- Note the
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
where a is a constant. - Hyperbolas have 2 asymptotes:
and .- 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
, then to , 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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