Question

Which one of the following represents the vertex form of a quadratic function?
(a) \[f\left( x \right)=a{{x}^{2}}+bx+c\]
(b) \[f\left( x \right)=a{{\left( x-h \right)}^{2}}+k\]
(c) \[f\left( x \right)=\left( x-{{x}_{1}} \right)\cdot \left( x-{{x}_{2}} \right)\]
(d) none of these

Answer 1

Hint: We start solving the problem by recalling the fact that the quadratic equation resembles the equation of parabola with axis parallel to y-axis. We then recall the equation of the parabola with vertex at $\left( h,k \right)$ and the axis parallel to y-axis is $a{{\left( x-h \right)}^{2}}=y-k$. We then make the necessary calculations to find the equation resembling $y=f\left( x \right)$. We then replace y with $f\left( x \right)$ to find the required form of the quadratic function.

Complete step by step solution:
According to the problem, we need to find the vertex form of a quadratic function.
We know that the quadratic equation resembles the equation of parabola with an axis parallel to the y-axis.
We know that the equation of the parabola with vertex at $\left( h,k \right)$ and axis parallel to y-axis is $a{{\left( x-h \right)}^{2}}=y-k$.
Now, we have $y=a{{\left( x-h \right)}^{2}}+k$.
As we can see that the polynomial on the right-hand side resembles the quadratic equation. So, let us assume $y=f\left( x \right)$.
$\Rightarrow f\left( x \right)=a{{\left( x-h \right)}^{2}}+k$, which is the vertex form of the required quadratic equation.
So, we have found the vertex form of a quadratic equation as $f\left( x \right)=a{{\left( x-h \right)}^{2}}+k$.

So, the correct answer is “Option B”.

Note: We should know that the quadratic equation will be of the form $a{{x}^{2}}+bx+c=0$ which resembles the equation of parabola with axis parallel to zero when the value of ‘y’ is set equal to 0. We should that the equation $f\left( x \right)=\left( x-{{x}_{1}} \right).\left( x-{{x}_{2}} \right)$ is known as the two-point form of the quadratic function. Since the given problem has asked about the quadratic function not about the quadratic equation, we do not equate it to zero.