Parabola

One of the following two methods can be chosen:

$X=\frac{-b}{2a}$

The value of $X$ that we receive will be replaced in the parabola function and we will obtain the value of $Y$ relevant.

The formula to find $X$ a vertex using two symmetric points is:

The vertex $X$ that we receive in the function to find the value of the vertex $Y$.

Now, we will move on to the points of intersection of the parabola with the $X$ and $Y$ axes

When we want to find the point of intersection with the $X$ axis:

We will set $Y=0$ in the quadratic equation and solve using a trinomial or the root formula.

We can find parabolas that are not zero and that do not have any point of intersection with the $X$ axis, or that have $1$ or a maximum of $2$.

When we want to find a point of intersection with the $Y$ axis:

We will set $X=0$ in the quadratic equation and find the solutions.

Wonderful. Now we will move on to the areas of increase and decrease of the quadratic function.

The areas of increase and decrease describe the $X$ where the parabola increases and where the parabola decreases.

The parabola changes its domain once, at the vertex.

Let's see this in the figure:

When there is a graph:

We will examine what happens when the $X$s are smaller than the vertex $X$ and what happens when the $X$s are greater than the vertex $X$.

When there is no graph:

1. We will examine the equation of the function and determine based on the coefficient of $X^2$ whether it is a minimum or maximum function.
2. Find the vertex $X$ according to the formula or by symmetric points.
3. We will plot a graph according to the data we have found and clearly see the areas of increase and decrease.

Positive domain: describes the $X$ where the graph of the parabola is above the $X$ axis, with a $Y$ value positive.

Negative domain: describes the $X$ where the graph of the parabola is below the $X$ axis, with a negative $Y$ value.

To find the domains of positivity and negativity, we will plot the graph of the parabola and ask:

At what $X$ values is the graph of the parabola above the $X$ axis, with a positive $Y$ value? This will be the domain of positivity of the parabola.

At what $X$ values is the graph of the parabola below the $X$ axis, with a negative $Y$ value? This will be the domain of negativity of the parabola.

Let's see this on the graph:

We will find the points of intersection with the axes and mark them on the coordinate system.

1. Find the vertex of the parabola and mark it on the coordinate system.
2. We will understand if the parabola is a maximum or minimum (according to the coefficient $a$) and will draw accordingly.