f(x)=ax2+bx+c f\left(x\right)=ax^2+bx+c

Where a0 a\ne0 , since if the coefficient a 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(x)=8x22x+4 f\left(x\right)=8x^2-2x+4

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

Example 2:

f(x)=7x3+6x2+2x1 f\left(x\right)=-7x^3+6x^2+2x-1

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

Quadratic function

The equation of the basic quadratic function is: y=ax2+bx+cy=ax^2+bx+c

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

ax2 ax^2 is called the squared term, quadratic term, or second-degree term.

a a is the coefficient of the quadratic term.

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

b b is the coefficient of the linear term.

c c is the constant term.

x 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(x)=3x25x+2 f\left(x\right)=3x^2-5x+2

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

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


Practice The quadratic function

Exercise #1

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

3x2+8x5 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 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.

Answer

8

Exercise #2

y=x2+10x 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 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.

 

Answer

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

Exercise #3

y=2x25x+6 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 a=2,b=-5,c=6

Exercise #4

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

3x2+5x 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 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.

Answer

0

Exercise #5

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

x2+7x9 -x^2+7x-9

Video Solution

Answer

-1

Exercise #1

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

4x2+9x2 4x^2+9x-2

Video Solution

Answer

-2

Exercise #2

y=x2 y=x^2

Video Solution

Answer

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

Exercise #3

y=x26x+4 y=x^2-6x+4

Video Solution

Answer

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

Exercise #4

y=2x23x6 y=2x^2-3x-6

Video Solution

Answer

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

Exercise #5

y=2x2+3x+10 y=-2x^2+3x+10

Video Solution

Answer

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

Exercise #1

y=3x2+4x+5 y=3x^2+4x+5

Video Solution

Answer

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

Exercise #2

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

x2=2x+7 x^2=2x+7

Video Solution

Answer

∓2

Exercise #3

y=x2+x+5 y=x^2+x+5

Video Solution

Answer

a=1,b=1,c=5 a=1,b=1,c=5

Exercise #4

y=x2+x+5 y=-x^2+x+5

Video Solution

Answer

a=1,b=1,c=5 a=-1,b=1,c=5

Exercise #5

y=4+3x2x y=4+3x^2-x

Video Solution

Answer

a=3,b=1,c=4 a=3,b=-1,c=4