Question

Find the “\[y\]” coordinate of the vertex of the parabola?

Answer 1

Hint:Parabola is the trace of points such that distance of any point on that trace is equal to a fixed point known as focus and coordinate is known as focal coordinate, and also every point has a fixed distance from a straight line known as directrix.
Equation of parabola can be given as: \[{y^2} = 4ax\]

Complete step by step solution:
For any quadratic equation say \[a{x^2} + bx + c = y\]

Vertex can be defined as putting x-coordinate as “\[h\]” and then assume the other coordinate as zero and since in the equation of the parabola starting from \[(0,0)\] the constant value in the quadratic equation will be zero.

We know for the vertex \[(h,k)\]

The first vertex \[h = \dfrac{{ - b}}{{2a}}\]

And then evaluating \[y\,at\,h\]

We get the second vertex \[k\]

You can also use \[k = \dfrac{{4ac - {b^2}}}{{2a}}\]

So the vertex of any parabola in general form can be written as,
\[(h,k) = (\dfrac{{ - b}}{{2a}},\dfrac{{4ac - {b^2}}}{{2a}})\]

Formulae Used: \[(h,k) = (\dfrac{{ - b}}{{2a}},\dfrac{{4ac - {b^2}}}{{2a}})\],\[{y^2} = 4ax\] and \[a{x^2}
+ bx + c = y\]

Additional Information: For finding the vertex of the parabola you should know at least one the vertex formulae to know the other, or you can just assume one coordinate as zero and now get the other coordinate you will get the same value.

Note: For any parabola its equation decides the opening of mouth of parabola, the axis coordinate which have the square term in equation will have opening towards itself, parabola also known as smiling and sad face, when the parabola is created in negative axis then it is known as sad face, whereas when parabola is created in positive axis then it is known as smiling face.