**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?

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

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

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

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

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

Question 2

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

Question 3

\( y=x^2 \)

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

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

Question 2

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

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.

What is the value of the coefficient $b$ in the equation below?

$3x^2+8x-5$

The quadratic equation of the problem is already arranged (that is, all the terms on one side and 0 on the other side), so we approach answering the question posed:

**In the problem, the question was asked: **what is the value of the coefficient$b$in the equation?

**Let's remember** the definitions of the coefficients when solving a quadratic equation and the formula for the roots:

The rule says that the roots of an equation of the form

$ax^2+bx+c=0$__are :__

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

**That is **the coefficient$b$is the coefficient of the **term in the first power** -$x$**We examine **the equation of the problem:

$3x^2+8x-5 =0$That is, the number that multiplies

$x$ is

$8$And then we recognize b, which is the coefficient of the term in the first power, is the number$8$,

__The correct answer is option d.__

8

$y=x^2+10x$

Here we have a quadratic equation.

A quadratic equation is always constructed like this:

$y = ax²+bx+c$

Where a, b, and c are generally already known to us, and the X and Y points need to be discovered.

Firstly, it seems that in this formula we do not have the C,

Therefore, we understand it is equal to 0.

$c = 0$

a is the coefficient of X², here it does not have a coefficient, therefore

$a = 1$

$b= 10$

is the number that comes before the X that is not squared.

$a=1,b=10,c=0$

$y=2x^2-5x+6$

In fact, a quadratic equation is composed as follows:

y = ax²-bx-c

That is,

a is the coefficient of x², in this case 2.

b is the coefficient of x, in this case 5.

And c is the number without a variable at the end, in this case 6.

$a=2,b=-5,c=6$

What is the value of the coefficient $c$ in the equation below?

$3x^2+5x$

The quadratic equation of the problem is already ordered (that is, all the terms on one side and 0 on the other side), so we approach answering the question posed:

**In the problem, the question was asked: **what is the value of the coefficient$c$in the equation?

**Let's remember** the definitions of the coefficients when solving a quadratic equation and the formula for the roots:

The rule says that the roots of an equation of the form

$ax^2+bx+c=0$__are:__

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

**That is **the coefficient

$c$is the free term - that is, the coefficient of the **term raised to the power of zero** -$x^0$(And this is because any number other than zero raised to the power of zero equals 1:

$x^0=1$)

**We examine **the equation of the problem:

$3x^2+5x=0$Note that there is no free term in the equation, that is, the numerical value of the free term is 0, in fact the equation can be written as follows:

$3x^2+5x+0=0$and therefore the value of the coefficient$c$ is 0.

__The correct answer is option c.__

0

Check your understanding

Question 1

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

Question 2

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

Question 3

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

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

Related Subjects

- Symmetry in a parabola
- Plotting the graph of the quadratic function and examining the roles of the parameters a, b, c in the function of the form y = ax^2 + bx + c
- Finding the zeros of a parabola
- Quadratic Inequality
- Methods for solving a quadratic function
- 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
- Quadratic Equations System - Algebraic and Graphical Solution
- Solution of a system of equations - one of them is linear and the other quadratic
- Intersection between two parabolas
- Squared Trinomial
- Word Problems
- Properties of the roots of quadratic equations - Vieta's formulas
- Ways to represent a quadratic function
- Families of Parabolas