**The Parabola** **$y=ax^2+bx+c$**

This function is a quadratic function and is called a parabola.

We will focus on two main types of parabolas: maximum and minimum parabolas.

This function is a quadratic function and is called a parabola.

We will focus on two main types of parabolas: maximum and minimum parabolas.

Also called smiling or happy.

A vertex is the minimum point of the function, where $Y$ is the lowest.

We can identify that it is a minimum parabola if the $a$ equation is positive.

Also called sad or crying.

A vertex is the maximum point of the function, where $Y$ is the highest.

We can identify that it is a maximum parabola if the $a$ equation is negative.

To the parabola,

the vertex marks its highest point.

How do we find it?

\( y=x^2 \)

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.

Test your knowledge

Question 1

\( y=x^2+10x \)

Question 2

\( y=x^2-6x+4 \)

Question 3

\( y=2x^2-5x+6 \)

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.**

Do you know what the answer is?

Question 1

\( y=2x^2-3x-6 \)

Question 2

\( y=-2x^2+3x+10 \)

Question 3

\( y=3x^2+4x+5 \)

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:**

- 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.
- Find the vertex $X$ according to the formula or by symmetric points.
- 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:**

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

Check your understanding

Question 1

What is the value ofl coeficiente \( a \) in the equation?

\( -x^2+7x-9 \)

Question 2

What is the value of the coefficient \( b \) in the equation below?

\( 3x^2+8x-5 \)

Question 3

What is the value of the coefficient \( c \) in the equation below?

\( 4x^2+9x-2 \)

Related Subjects

- Symmetry in a parabola
- The functions y=x²
- Family of Parabolas y=x²+c: Vertical Shift
- Family of Parabolas y=(x-p)²
- Family of Parabolas y=(x-p)²+k (combination of horizontal and vertical shifts)
- Standard Form of the Quadratic Function
- Vertex form of the quadratic equation
- Factored form of the quadratic function
- Completing the square in a quadratic equation
- Various Forms of the Quadratic Function
- Quadratice Equations and Systems of Quadraric Equations
- Squared Trinomial
- Word Problems