In algebra when we talk about symmetry, we are talking about a line that divides a figure exactly in half, so when we are working with quadratic functions, we had already mentioned that its graph will always be a parabola, therefore the axis of symmetry will be the line that divides the parabola in half, it's as if that axis were a mirror and the part on the right side reflected on the left side and vice versa.

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The functions y=x²

In the general form of a quadratic function, we mention that there are three terms: the quadratic term, linear term, and the constant term, to which we can say it is a complete quadratic function for having all its terms, however, this will not always be the case, that is we can have an incomplete quadratic function where it may lack the linear term or the constant term or even both terms, an incomplete function of the form $y=x^2$ means that $b=0$ and $c=0$. This is the most basic form of a quadratic function and its graph will be a parabola with the vertex at the origin of the Cartesian plane, that is, at $\left(0,0\right)$.

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Test your knowledge

Question 1

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

In the case of the family of parabolas $y=x^2+C$, we now have the quadratic term and the constant term, graphically the constant term, that is $C$, will indicate whether the graph of the parabola shifts up or down on the $Y$ axis depending on the value of $C$.

Unlike the previous parabolas, now the term $p$ will indicate that the parabola will move horizontally, only if $p>0$ then the parabola will move to the left (negative $X$ axis) the value of $p$; if $p<0$, that is, negative, it will now move to the right (positive $X$ axis).

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Family of parabolas y=(x-p)²+k (combination of horizontal and vertical shifts)

This family of parabolas is the combination of the two previous ones, that is; The parabola will also move up or down (vertically) depending on the value of $K$. And $p$ will indicate if it moves to the right or left (horizontally).

As we well know, factoring is writing a term as a multiplication, in the case of a quadratic function it is similar, the factored form will be written as a multiplication and in this case it helps us find the intersections of the parabola on the $X$ axis. In other words, we will find the points where the parabola intersects with the $X$ axis. The factored form can be observed as follows:

Finding the zeros of a quadratic function through its form

A quadratic function can be written in the following way:

$f\left(x\right)=ax^2+bx+c=0$

We can observe that the function is equal to zero, therefore we can find the zeros, graphically they will be the intersections of the parabola with the $X$ axis. There are various ways to find these zeros, one of them will be by the factoring method or by the quadratic formula method.

Completing the square in a quadratic equation

We already mentioned that a quadratic function can be in complete or incomplete form, the latter being those when it lacks one of the terms, either the linear term or the independent term, but never the quadratic term, since without this term it is no longer a quadratic function.

When we have an incomplete equation, in the case where we only have the quadratic term and linear term, we can complete the square through a series of steps. This is, find a number in such a way that we have all three terms and that it can be factored by the perfect square trinomial and thus find the solutions to the equation.

To find the solution of a quadratic equation, we can find it by factorization or by the quadratic formula, also called the general formula. This formula will help us find the solutions or roots that satisfy the equality in the equation. Graphically, we will find where it cuts the $X$ axis or the intersections of the parabola with said axis. The coefficients of the terms that make up the quadratic function tell us certain things in the graph of the parabola.

From the quadratic equation $ax^2+bx+c=0$

$a$, $b$, and $c$ are coefficients or in this case, we will call them parameters. The quadratic formula that will help us find the roots is:

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

A quadratic equation can have at most two solutions.

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Do you know what the answer is?

Question 1

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

Equations and systems of equations of second degree or quadratic

Just as in the case of a linear equation we encounter systems of linear equations where we study various methods to find the solution to the system, in the case of quadratic equations we can also work with systems of quadratic equations where we must also find the values of $X$ and $Y$ in such a way that the solutions satisfy both equations. Graphically, this system of equations can be observed as two parabolas where we find the coordinates of the two intersection points in the case of having two solutions, one intersection point when having a single solution, and lastly no intersection point when the system has no solution.

Quadratic Equations System - Algebraic and Graphical Solution

In a system of quadratic equations, we can find its solution algebraically and also graphically as we did with systems of linear equations. When we find the solution to a system of quadratic equations by the algebraic method, we can do it by equating, while graphically we can observe it with the points of intersection between the two parabolas, and we say that two systems of equations:

Have a unique solution when the parabolas only intersect at a single point.

Have two solutions when it can be observed that there are two points of intersection

And there is no solution when there is no point of intersection.

Solution of a system of equations when one of them is linear and the other quadratic

We may encounter systems of two equations where one of them is linear and the other is quadratic. We will solve this type of systems by the substitution method, where we will solve for the variable $y$ in the linear equation and substitute this value into the quadratic, and then solve the resulting quadratic equation using the general formula. The solution or solutions will be the intersections between a straight line and a parabola.

As we previously mentioned, we can encounter three types of cases when we want to find the solution to a system of two quadratic equations. When these two equations are graphed on the Cartesian plane, it will guide us to find the solution:

There is a unique solution when the parabolas intersect at only one point.

There are two solutions when it can be observed that there are two points of intersection

And there is no solution when there is no point of intersection.

When we have problem applications that involve quadratic equations, we must use algebraic language, What does this mean? When we are presented with these types of problems, we must formulate our equations using the statements provided to us in such a way that we must translate this normal language into algebraic language and structure the equations, in this case, two equations to form our system of equations and subsequently find the solution by any method studied so far, and once the solution to the system of equations is obtained, now find the correct and viable solution to the initial problem.

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Quadratic Inequality

Now we will talk about an inequality, that is, something that is not equal, and in this case, we will not have one or two solutions, but rather a solution set, where the solution will indicate in which range of solutions will make a quadratic equation positive or negative. In the following way:

$ax^2+bx+c<0$

$ax^2+bx+c>0$

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Test your knowledge

Question 1

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

A quadratic function can be presented in the following ways:

Algebraically: $f\left(x\right)=ax^2+bx+c$

Numerically: tabulation

Graphically: Parabola.

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Ways to Represent a Function

A function can be presented in four ways:

Verbally

Algebraically (Relation function)

Numerically (tabulation)

Graphically

Examples and exercises with solutions of the quadratic function

Exercise #1

$y=x^2+10x$

Video Solution

Step-by-Step Solution

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.

Answer

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

Exercise #2

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

Video Solution

Step-by-Step Solution

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.

Answer

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

Exercise #3

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

$3x^2+8x-5$

Video Solution

Step-by-Step Solution

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.

Answer

8

Exercise #4

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

$3x^2+5x$

Video Solution

Step-by-Step Solution

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.

Answer

0

Exercise #5

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

$4x^2+9x-2$

Video Solution

Answer

-2

Do you know what the answer is?

Question 1

Create an algebraic expression based on the following parameters: