Question

As part of the 50th Anniversary in the gymnasium, you and your crew will unveil a new, specially designed banner from the roof directly above the centre of the gym floor. Upon closer inspection, you notice that the bottom portion of the banner is in the shape of a parabola Determine the equation of the parabola, in root form, based on the graph above.

Answer 1

Solution:

Given:

Based on the graph, it can be estimated that;

The vertex is at (0,-3) and the roots are at (-5,0) and (5,0)

Using the equation of a parabola in vertex form;

$y=a(x-h)^2+k$

where;

$\begin{gathered} (h,k)\text{ is the vertex} \\ h=0 \\ k=-3 \\ \\ Hence, \\ y=a(x-0)^2+(-3) \\ y=ax^2-3 \end{gathered}$

Using the point (5,0) to get the constant a;

$\begin{gathered} x=5 \\ y=0 \\ \\ Hence,\text{ }y=ax^2-3 \\ 0=a(5^2)-3 \\ 3=25a \\ \frac{3}{25}=a \\ a=\frac{3}{25} \end{gathered}$

Hence, the equation of the parabola is;

$\begin{gathered} y=ax^2-3 \\ y=\frac{3}{25}x^2-3 \end{gathered}$