Parabola

🏆Practice the function y=ax^2+bx+c

The Parabola y=ax2+bx+c 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.

Minimum Parabola

Also called smiling or happy.

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

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

1b - We can identify that it is a minimum parabola if the equation a is positive


Maximum Parabola

Also called sad or crying.

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

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

2b - 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?


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Test yourself on the function y=ax^2+bx+c!

einstein

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

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

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Find the vertex of the parabola

One of the following two methods can be chosen:


The first method: using the formula for the vertex of the parabola

X=b2aX=\frac{-b}{2a}

The value of XX that we receive will be replaced in the parabola function and we will obtain the value of YY relevant.


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The second method: using two symmetric points

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

The vertex XX that we receive in the function to find the value of the vertex YY.

Now, we will move on to the points of intersection of the parabola with the XX and YY axes


Point of Intersection with the Axes

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

We will set Y=0Y=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 XX axis, or that have 11 or a maximum of 22.

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

We will set X=0X=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?

Areas of Increase and Decrease

The areas of increase and decrease describe the XX 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 XXs are smaller than the vertex XX and what happens when the XXs are greater than the vertex XX.

When there is no graph:

  1. We will examine the equation of the function and determine based on the coefficient of X2X^2 whether it is a minimum or maximum function.
  2. Find the vertex XX 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 and Negative Domains

Positive domain: describes the XX where the graph of the parabola is above the XX axis, with a YY value positive.

Negative domain: describes the XX where the graph of the parabola is below the XX axis, with a negative YY value.

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

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

At what XX values is the graph of the parabola below the XX axis, with a negative YY 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 aa) and will draw accordingly.

Examples and exercises with solutions of parabola

examples.example_title

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

3x2+8x5 3x^2+8x-5

examples.explanation_title

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 coefficientb 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

ax2+bx+c=0 ax^2+bx+c=0 are :

x1,2=b±b24ac2a x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

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

3x2+8x5=0 3x^2+8x-5 =0 That is, the number that multiplies

x x is

8 8 And then we recognize b, which is the coefficient of the term in the first power, is the number8 8 ,

The correct answer is option d.

examples.solution_title

8

examples.example_title

y=x2+10x y=x^2+10x

examples.explanation_title

Here we have a quadratic equation.

A quadratic equation is always constructed like this:

 

y=ax2+bx+c 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 c = 0

 

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

a=1 a = 1

 

b=10 b= 10

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

 

examples.solution_title

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

examples.example_title

y=2x25x+6 y=2x^2-5x+6

examples.explanation_title

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.

examples.solution_title

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

examples.example_title

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

3x2+5x 3x^2+5x

examples.explanation_title

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 coefficientc 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

ax2+bx+c=0 ax^2+bx+c=0 are:

x1,2=b±b24ac2a x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

That is the coefficient
c c is the free term - that is, the coefficient of the term raised to the power of zero -x0 x^0 (And this is because any number other than zero raised to the power of zero equals 1:

x0=1 x^0=1 )

We examine the equation of the problem:

3x2+5x=0 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:

3x2+5x+0=0 3x^2+5x+0=0 and therefore the value of the coefficientc c is 0.

The correct answer is option c.

examples.solution_title

0

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