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General quadratic function
Definition
The general quadratic function has the following form: \[ f(x) = A(x-h)^2 + k, \] where $x$ is the input and $A,h$ and $k$ are the parameters.
Parameters
- $A$: slope multiplier.
- The larger the value of $A$ the steeper the slope.
- If $A<0$ (negative), then the function opens downwards.
- $h$: horizontal displacement of the function centre. Notice that subtracting a
number inside the bracket $(\ )^2$ (i.e. positive $h$) makes the function go to the right.
- $k$: vertical displacement of the function.
If a quadratic crosses the $x$-axis, then it can be written in factored form \[ f(x) = (x-a)(x-b), \] where $a$ and $b$ are the two roots.
Another very common way of writing a quadratic function is \[ f(x) = Ax^2 + Bx + C. \]
Properties
- There is a unique parabola that passes through any three points $(x_1,y_1),$ $(x_2,y_2)$ and $(x_3,y_3)$ if
the points have different $x$ coordinates $x_1 \neq x_2$, $x_2 \neq x_3$ and $x_1 \neq x_3$.
- The derivative of $f(x)=Ax^2 + Bx + C$ is $f'(x)=2Ax + B$.
Graph
When $h=1$ (one unit shifted to the right) and $k=-2$ (two units shifted downwards), we get the following graph:
