Vertex of a parabola

In this article, we will study the vertex of the parabola and discover easy ways to find it without too much effort.
The vertex of the parabola marks the highest point of a sad-faced parabola and, the lowest point of a happy-faced parabola.
Let's remember the equation of the parabola:
$y=ax^2+bx+c$

Reminder: 
$a$  positive –> happy-faced parabola
$a$  negative –> sad-faced parabola  

The notation of the parabola's vertex is as follows: $(Y~vertex, X~vertex)$

To find the vertex of the parabola, we must solve for the value of its $X$ and its $Y$.

To find the value of the $X$ of the vertex:
We will use the following formula: $x=\frac{(-b)}{2a}$

To find the value of the $Y$ of the vertex:
We will place the value of the $X$ we have found into the original parabola equation and obtain the $Y$ of the vertex.

Here is the following equation of the parabola –>
$y=5x^2+20x+4$
Find the vertex of the parabola.

Solution:
To find the $X$ of the vertex we will place it in the formula $x=\frac{(-b)}{2a}$
$b=2$
$a=5$
We will obtain:
$x=\frac{-20}{2 \times 5}$
$x=\frac{-20}{10}$
$x=-2$

To find the $Y$ of the vertex we will place the $X$ of the vertex we have found: $-2$
in the original parabola equation.
We will obtain:
$y=5 \times (-2)^2+20 \times (-2)+4$
$y=5 \times 4-40+4$
$y=20-40+4$
$y=-16$

The vertex of the parabola is $(-2,-16)$

Note: The fact of having obtained a parabola vertex with negative numbers does not mean that the parabola is a sad face parabola.

To find the vertex of the parabola in this way, we must first find the points of intersection of the parabola with the $X$ axis.
To do this, we will set $Y=0$ in the original parabola equation, solve the quadratic equation with the help of the quadratic formula, and obtain two values of $X$.
Reminder: The quadratic formula to solve a quadratic equation is: $x = {-b \pm \sqrt{b^2-4ac} \over 2a}$

Then, we will find the point that is exactly between the two $X$ values we obtained, and that will be the $X$ of the vertex.
To find the midpoint, we will calculate the average of the $X$s.

After finding the $X$ of the vertex, we will place it in the original parabola equation and obtain the $Y$ of the vertex.

Here is the following parabola equation ->
$f(x)=x^2-8x+12$
Find the vertex of the parabola.

First step: Find the points of intersection with the $X$ axis

We will set $Y=0$
We will obtain:
$x^2-8x+12=0$
We will solve the quadratic equation by placing the data in the quadratic formula and we will obtain:

$x_{1,2} = {-8 \pm \sqrt{8^2-4 \times 1 \times 12} \over 2 \times 1}$

$x_{1,2} = {-8 \pm 4 \over 2}$

$X_1=6$
$X_2=2$

Second step: Calculating the mean we will find the point that is exactly between the two intersection points –> $X$ of the vertex.

We will obtain:
$X={(6+2)\over2}=4$

Third step: We will place the $X$ of the vertex we obtained into the original parabola equation and find the $Y$ of the vertex.

We will obtain:

$Y=4^2-8 \times 4+12$
$Y=-4$

The vertex of the parabola is:
$(4,-4)$

As you can see, the second method seems to be quite longer.
However, if you already have $2$ points of intersection of the parabola with the $X$ axis, it is advisable to use this method, find the point that is exactly between them by calculating the average and continue looking for the $Y$ of the vertex by placing the data in the original equation.