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

Where $a\ne0$, since if the coefficient $a$ does not appear then it would not be a quadratic function.

The graph of a quadratic function will always be a parabola.

Example 1:

$f\left(x\right)=8x^2-2x+4$

It is a quadratic or second-degree function because its largest exponent is $2$.

Example 2:

$f\left(x\right)=-7x^3+6x^2+2x-1$

This is not a second-degree function because, although it has an exponent $2$, its largest exponent is $3$.

The equation of the basic quadratic function is: $y=ax^2+bx+c$

This way of writing them is called the general form of a second-degree function, where:

$ax^2$ is called the squared term, quadratic term, or second-degree term.

$a$ is the coefficient of the quadratic term.

$bx$ is called the linear term or first-degree term.

$b$ is the coefficient of the linear term.

$c$ is the constant term.

$x$ is called the unknown of the function and represents the number or numbers that make the function or in this case the equation true.

Example:

$f\left(x\right)=3x^2-5x+2$

$a=3$, $b=-5$ and $c=2$

We must remember that for an equation to be of second degree, $a$ must always be different from zero.

## Examples with solutions for The Quadratic Function

### Exercise #1

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

$3x^2+8x-5$

### Step-by-Step Solution

The quadratic equation of the given problem is already arranged (that is, all the terms are found on one side and the 0 on the other side), thus we approach the given problem as follows;

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

Let's remember the definitions of coefficients when solving a quadratic equation as well as 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 then examine the equation of the given problem:

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

$x$ is

$8$Consequently we are able to identify b, which is the coefficient of the term in the first power, as the number$8$,

Thus the correct answer is option d.

8

### Exercise #2

$y=x^2+10x$

### 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.

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

### Exercise #3

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

### 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.

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

### Exercise #4

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

$3x^2+5x$

### Step-by-Step Solution

The quadratic equation of the given problem has already been arranged (that is, all the terms are on one side and 0 is on the other side) thus we can approach the question as follows:

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

Let's remember the definition of a coefficient when solving a quadratic equation as well as 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 - and as such the coefficient of the term is raised to the power of zero -$x^0$(Any number other than zero raised to the power of zero equals 1:

$x^0=1$)

Next we examine the equation of the given 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.

Hence the correct answer is option c.

0

### Exercise #5

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

$4x^2+9x-2$

-2

### Exercise #6

$y=-2x^2+3x+10$

### Video Solution

$a=-2,b=3,c=10$

### Exercise #7

$y=x^2$

### Video Solution

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

### Exercise #8

$y=x^2-6x+4$

### Video Solution

$a=1,b=-6,c=4$

### Exercise #9

$y=3x^2+4x+5$

### Video Solution

$a=3,b=4,c=5$

### Exercise #10

$y=2x^2-3x-6$

### Video Solution

$a=2,b=-3,c=-6$

### Exercise #11

What is the value ofl coeficiente $a$ in the equation?

$-x^2+7x-9$

-1

### Exercise #12

Create an algebraic expression based on the following parameters:

$a=2,b=2,c=2$

### Video Solution

$2x^2+2x+2$

### Exercise #13

Create an algebraic expression based on the following parameters:

$a=2,b=\frac{1}{2},c=4$

### Video Solution

$2x^2+\frac{1}{2}x+4$

### Exercise #14

Create an algebraic expression based on the following parameters:

$a=2,b=0,c=4$

### Video Solution

$2x^2+4$

### Exercise #15

Create an algebraic expression based on the following parameters:

$a=2,b=4,c=8$

### Video Solution

$2x^2+4x+8$