Question

What is the following parabola’s axis of symmetry? $y = {\left( {x - 1} \right)^2} + 1$
A. $x = 1$
B. $y = 1$
C. $x = - 1$
D. None of these

Answer 1

Hint:
Compare the given equation of the parabola with the standard equation of the parabola, $4p\left( {y - k} \right) = {\left( {x - h} \right)^2}$, where the vertex of the equation is $\left( {h,k} \right)$ and the coordinates of focus of the parabola is $\left( {h,k + p} \right)$ and determine the vertex of the parabola. The vertex will indicate the shifting of parabola from the origin, which will also give the axis of symmetry of parabola.

Complete step by step solution:
We are given that the equation of parabola is $y = {\left( {x - 1} \right)^2} + 1$
As it is known that the standard equation of parabola is $4p\left( {y - k} \right) = {\left( {x - h} \right)^2}$, where the vertex of the equation is $\left( {h,k} \right)$ and the coordinates of focus of the parabola is $\left( {h,k + p} \right)$
Compare the given equation $y = {\left( {x - 1} \right)^2} + 1$ with the standard equation of the parabola, $4p\left( {y - k} \right) = {\left( {x - h} \right)^2}$.
Then, the vertex of the parabola is $\left( {1,1} \right)$
This implies that the equation of a parabola is shifted to the right.
Also, the axis of symmetry is the line that divides the parabola into two equal parts.
If the equation had been ${y^2} = 4ax$, the axis of symmetry was $y$ axis , that is $x = 0$
But after shifting it to right, the axis of symmetry is $x = 1$

Hence, option A is correct.

Note:
A parabola is a locus of points which are at equal distance from a fixed point called focus and a fixed-line called the directrix. The parabola whose equation is ${y^2} = 4ax$ is symmetric to $y$ axis whereas the parabola with the equation ${x^2} = 4ay$ is symmetric to $x$ axis.