Question

How do you write the equation for the quadratic function with roots $0$ and $2$ and a vertex at $\left( 1,5 \right)$ ?

Answer 1

Hint: Here in this question we have been asked to write the equation for the quadratic function with the roots 0 and 2 and having the vertex at $\left( 1,5 \right)$ . From the basic concepts of quadratic equations we know the general form of a quadratic equation is given as $a{{x}^{2}}+bx+c=y$ .

Complete step by step answer:
Now considering from the question we have been asked to write the equation for the quadratic function with the roots 0 and 2 and having the vertex at $\left( 1,5 \right)$ .
From the basic concepts of quadratic equations we know the general form of a quadratic equation is given as $a{{x}^{2}}+bx+c=y$ .
Now as we have the roots of the expression we can say that
 $\begin{align}
  & \Rightarrow \left( x-0 \right)\left( x-2 \right)=a{{x}^{2}}+bx+c \\
 & \Rightarrow {{x}^{2}}-2x \\
\end{align}$
Now as the vertex $\left( 1,5 \right)$ should also satisfy the quadratic function. We need to have a constant factor mathematically given as $y=k\left( x-{{x}_{1}} \right)\left( x-{{x}_{2}} \right)$ where ${{x}_{1}},{{x}_{2}}$ are the roots and $k$ is the constant factor.
Hence we will have
 $\begin{align}
  & 5=k\left( 1-2 \right) \\
 & \Rightarrow k=-5 \\
\end{align}$ .

Therefore we can conclude that the equation for the quadratic function with the roots 0 and 2 and having the vertex at $\left( 1,5 \right)$ is given as $y=-5{{x}^{2}}+10x$.

Note: During the process of answering questions of this type we should be sure with the quadratic equations concepts and the calculations that we are going to apply and perform in between the steps. This is a very simple and easy question and can be answered accurately in a short span of time. Very few mistakes are possible in questions of this type. Someone can forget about the concept of vertex and write the answer as $y={{x}^{2}}-2x$ which will be a wrong answer.