The Quadratic Function - Examples, Exercises and Solutions

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

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 $3$ .

Quadratic function

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.

Detailed explanation

Practice The Quadratic Function

Question 1

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

Question 2

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

Question 3

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

$$ 3x^2+8x-5 $$

Question 4

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

$$ 3x^2+5x $$

Question 5

$$ y=x^2 $$

Examples with solutions for 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 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.

Answer

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 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.

Answer

Exercise #5

$y=x^2$

Video Solution

Answer

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

Question 1

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

Question 2

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

Question 3

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

Question 4

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

Question 5

Answer

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

Exercise #10

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

$-x^2+7x-9$

Video Solution

Answer

-1

Question 1

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

$$ 4x^2+9x-2 $$

Question 2

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

Question 3

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

Question 4

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

Answer

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