1 Quadratic functions

1.1 Quadratic functions and parabolas

Graphs of $y$ against $x$ resulting from quadratic functions ( HELM booklet  2.8, Table 1) are called parabolas . These take the general form: $y=a{x}^2+bx+c$ . The coefficients $a,b$ and $c$ influence the shape, form and position of the graph of the associated parabola. They are the parameters of the parabola. In particular the magnitude of $a$ determines how wide the parabola opens (large $a$ implies a narrow parabola, small $a$ implies a wide parabola) and the sign of $a$ determines whether the parabola has a lowest point (minimum) or highest point (maximum). Negative $a$ implies a parabola with a highest point. The most useful form of equation for determing the graphical appearance of a parabola is $y-C=A{\left(x-B\right)}^2$ . To see the relation between this form and the general form simply expand:

$y=A{x}^2-2ABx+A{B}^2+C$

so, comparing with $y=a{x}^2+bx+c$ we have:

$a\equiv A,b\equiv -2ABc\equiv A{B}^2+C$

We deduce that the relation between the two sets of constants $A,B,C$ and $a,b,c$ is:

$A=aB=-\frac{b}{2a}\text{and}C=c-\frac{b^2}{4a}$

This new form for the parabola enables the coordinates of the highest or lowest point, known as the vertex to be written down immediately. The coordinates of the vertex are given by $\left(B,C\right)$ . Changing the value of $B$ shifts the vertex, and hence the whole parabola, up or down. Changing the value of $C$ shifts the vertex, and hence the whole parabola, to left or right.

Task!

Assume the variation of an object’s location with time is represented by a quadratic function:-

$s=\frac{t^2}{9}\left(0\le t\le 30\right)$

Compare this function with the general form $y-C=A{\left(x-B\right)}^2$ .

1. What variables correspond to $y$ and $x$ in this case?
2. What are the values of $C,A$ and $B$ ?

Answer