Question

3. Is it possible to determine the equation of a quadratic function given the vertex and one other point on the parabola? Explain why or why not.

Answer 1

A parabola can be written in its vertex form, which is shown below

$\begin{gathered} y=a(x-h)^2+k \\ (h,k)\to\text{vertex} \end{gathered}$

Suppose that we know the vertex (h,k) and one other point on the parabola (b,c); then,

$\begin{gathered} y=a(x-h)^2+k \\ a\to to\text{ be defined} \end{gathered}$

We only need to know the value of a to completely identify the equation of the parabola, which can be done as shown below

$\begin{gathered} c=a(b-h^{})^2+k \\ \Rightarrow\frac{c-k}{(b-h^{})^2}=a \end{gathered}$

Then, we can indeed determine the equation given the vertex and one other point on the parabola.