Further Functions

Cubic Equations

 
A cubic equation adopts the form ax^3 + bx^2 + cx + d, where a =/= 0
If a > 0, the graph will look like this:
 Positive Cubic
If a < 0, the graph will look like this:
Negative Cubic
ax^3 + bx^2 + cx + d can also be arranged into the form a(x-h)^3 + k, and hence helps you to determine the translations.
NOTE:
 
If h > 0, the curve moves h units to the left
If h < 0, the curve moves h units to the right
If k > 0, the curve moves k units up
If k < 0, the curve moves k units down
Inverse Functions
 
An inverse function, as the name suggests, is the inverse of the original function. The domain of the original function will be the range of the inverse function; the range of the original function will be the domain of the inverse function.
While the function is denoted by f(x), inverse function is denoted by ƒ−1(x). In order to find the inverse of a function, you simply make the independent variable “x” the subject.
 
For example, the inverse of y = mx + c will be
(y – c)/m = x
 
Because we always want “x” to be the independent variable, we swap the x and y, the inverse function will therefore be
y = (x – c)/m
ƒ−1(x) = (x – c)/m
 
Please note that an inverse function is a reflection of the original function in y=x, meaning they are symmetrical about the line y = x.
 
Tangent
 
Suppose you have a parabola, and you want to find the gradient of the parabola at a certain coordinate, you will then draw a straight line that touches that one and only point and calculate its gradient. This line is called the tangent. Tangent, therefore is a line that touches only one single point of the curve.
Tangent
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