Question

How do you find the vertex of a parabola using the axis of symmetry?

Answer 1

Hint:
Every parabola has an axis of symmetry and it is defined as the line that divides the graph into two perfect halves. Also we should be aware of the fact that the vertex occurs on the vertical line of symmetry, which is not affected by shifting up or down.

Complete step by step solution:
Let's consider a generic parabola of the form:
\[y = a{x^2} + bx + c\]
The vertex occurs on the vertical line of symmetry, which is not affected by shifting up or down.
So even if we subtract c to obtain the parabola, i.e.,\[y = a{x^2} + bx + c - c\]
It becomes equal to \[y = a{x^2} + bx\] which has the same axis of symmetry.
\[y = a{x^2} + bx\] can also be written as: \[y = x(ax + b)\]
We will factorize the obtained expression, \[y = x(ax + b)\]
On factoring, we see that the x-intercepts of this parabola occur at \[x = 0\] and \[x = - \dfrac{b}{a}\]
We know that the axis of symmetry is the line that divides the graph into two perfect halves.
Hence, the axis of symmetry will lie between the obtained two pints which are \[x = 0\] and \[x = - \dfrac{b}{a}\].

Therefore, the axis of symmetry is equivalent to \[x = - \dfrac{{2b}}{a}\].

Note:
The vertex occurs on the vertical line of symmetry, which is not affected by shifting up or down. Even after we subtract the intercept from the given parabola form, the newly obtained parabola will also have the same axis of symmetry. The axis of symmetry is the line that divides the graph into two perfect halves.