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

Examples with solutions for The Quadratic Function

Exercise #1

Determine the value of the coefficient a a in the following equation:

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

Video Solution

Step-by-Step Solution

The quadratic equation in the problem is already arranged (meaning all terms are on one side and 0 on the other side), so let's proceed to answer the question asked:

The question asked in the problem - What is the value of the coefficienta a in the equation?

Let's recall the definitions of coefficients in solving quadratic equations and the roots formula:

The rule states 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 a a is the coefficient of the quadratic term (meaning the term with the second power)- x2 x^2 Let's examine the equation in the problem:

x2+7x9=0 -x^2+7x-9 =0

Remember that the minus sign before the quadratic term means multiplication by: 1 -1 , therefore- we can write the equation as:

1x2+7x9=0 -1\cdot x^2+7x-9 =0

The number that multiplies the x2 x^2 , is 1 -1 hence we identify that the coefficient of the quadratic term is the number 1 -1 ,

Therefore the correct answer is A.

Answer

-1

Exercise #2

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 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 coefficientb 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

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 then examine the equation of the given problem:

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

x x is

8 8 Consequently we are able to identify b, which is the coefficient of the term in the first power, as the number8 8 ,

Thus the correct answer is option d.

Answer

8

Exercise #3

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

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

Video Solution

Step-by-Step Solution

The quadratic equation is given as 4x2+9x2 4x^2 + 9x - 2 . This equation is in the standard form of a quadratic equation, which is ax2+bx+c ax^2 + bx + c , where a a , b b , and c c are coefficients.

  • The term 4x2 4x^2 indicates that the coefficient a=4 a = 4 .
  • The term 9x 9x indicates that the coefficient b=9 b = 9 .
  • The constant term 2-2 indicates that the coefficient c=2 c = -2 .

From this analysis, we can see that the coefficient c c is 2-2.

Therefore, the value of the coefficient c c in the equation is 2-2.

Answer

-2

Exercise #4

y=x2 y=x^2

Video Solution

Step-by-Step Solution

To solve this problem, let's follow these steps:

  • Step 1: Recognize the standard form of a quadratic equation.
  • Step 2: Match the given function to the standard form.
  • Step 3: Identify each coefficient a a , b b , and c c .

Now, let's work through these steps:

Step 1: The standard form of a quadratic function is y=ax2+bx+c y = ax^2 + bx + c . Our goal is to identify a a , b b , and c c .

Step 2: We are given the function y=x2 y = x^2 . This can be aligned with the standard form as y=1x2+0x+0 y = 1 \cdot x^2 + 0 \cdot x + 0 .

Step 3: By comparing the given function y=x2 y = x^2 with the standard form, we can deduce:
- The coefficient of x2 x^2 is 1, so a=1 a = 1 .
- The linear term coefficient is missing, which implies b=0 b = 0 .
- There is no constant term, so c=0 c = 0 .

Therefore, the coefficients are a=1,b=0,c=0 a = 1, b = 0, c = 0 , corresponding to choice 1.

Answer

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

Exercise #5

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 #6

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

Video Solution

Step-by-Step Solution

To solve this problem, we'll clearly delineate the given expression and compare it to the standard quadratic form:

  • Step 1: Recognize the standard form of a quadratic equation as y=ax2+bx+c y = ax^2 + bx + c .
  • Step 2: Compare the given equation y=x26x+4 y = x^2 - 6x + 4 to the standard form.
  • Step 3: Identify coefficients:
    - The coefficient of x2 x^2 is a=1 a = 1 .
    - The coefficient of x x is b=6 b = -6 .
    - The constant term is c=4 c = 4 .

Therefore, the coefficients for the quadratic function y=x26x+4 y = x^2 - 6x + 4 are a=1 a = 1 , b=6 b = -6 , and c=4 c = 4 .

Among the provided choices, choice 3: a=1,b=6,c=4 a=1,b=-6,c=4 is the correct one.

Answer

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

Exercise #7

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 #8

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

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Identify the given quadratic function.
  • Match it with the standard form of a quadratic equation y=ax2+bx+cy = ax^2 + bx + c.
  • Extract the values of aa, bb, and cc directly from the comparison.

Now, let's work through each step:
Step 1: The given quadratic function is y=2x23x6y = 2x^2 - 3x - 6.
Step 2: The standard form of a quadratic equation is y=ax2+bx+cy = ax^2 + bx + c.
Step 3: By matching the given quadratic function with the standard form:

- The coefficient of x2x^2 is 22, so a=2a = 2.
- The coefficient of xx is 3-3, so b=3b = -3.
- The constant term is 6-6, so c=6c = -6.

Therefore, the solution to the problem is a=2a = 2, b=3b = -3, c=6c = -6.

Answer

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

Exercise #9

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

Video Solution

Step-by-Step Solution

Let's determine the coefficients for the quadratic function given by y=2x2+3x+10 y = -2x^2 + 3x + 10 .

  • Step 1: Identify a a .
    The coefficient of x2 x^2 is 2-2. Thus, a=2 a = -2 .
  • Step 2: Identify b b .
    The coefficient of x x is 33. Thus, b=3 b = 3 .
  • Step 3: Identify c c .
    The constant term is 1010. Thus, c=10 c = 10 .

Comparing these coefficients to the provided choices, the correct answer is:

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

Therefore, the correct choice is Choice 4.

Answer

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

Exercise #10

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

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the given quadratic function.
  • Step 2: Compare it to the standard form y=ax2+bx+c y = ax^2 + bx + c .
  • Step 3: Determine the values of a a , b b , and c c .

Now, let's work through each step:

Step 1: The problem gives us the quadratic function y=3x2+4x+5 y = 3x^2 + 4x + 5 .

Step 2: The standard form of a quadratic function is y=ax2+bx+c y = ax^2 + bx + c .

Step 3: By comparing y=3x2+4x+5 y = 3x^2 + 4x + 5 with y=ax2+bx+c y = ax^2 + bx + c , we find:
- The coefficient of x2 x^2 is a=3 a = 3 .
- The coefficient of x x is b=4 b = 4 .
- The constant term is c=5 c = 5 .

Therefore, the solution to the problem is a=3,b=4,c=5 a = 3, b = 4, c = 5 .

This matches choice 2, which states: a=3,b=4,c=5 a = 3, b = 4, c = 5 .

Answer

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

Exercise #11

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 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 coefficientc 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

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 - and as such the coefficient of the term is raised to the power of zero -x0 x^0 (Any number other than zero raised to the power of zero equals 1:

x0=1 x^0=1 )

Next we examine the equation of the given 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.

Hence the correct answer is option c.

Answer

0

Exercise #12

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

Video Solution

Step-by-Step Solution

To solve this problem, we will identify the parameters of the given quadratic function step-by-step:

  • Step 1: Define the problem statement: We have y=x2+x+5 y = x^2 + x + 5 .
  • Step 2: Identify the standard form of a quadratic function, which is y=ax2+bx+c y = ax^2 + bx + c .
  • Step 3: Compare the given quadratic expression with the standard form to identify the coefficients.

Now, let's analyze the quadratic function provided:

From the given expression y=x2+x+5 y = x^2 + x + 5 :
- The coefficient of x2 x^2 is 1 1 , so a=1 a = 1 .
- The coefficient of x x is 1 1 , so b=1 b = 1 .
- The constant term is 5 5 , so c=5 c = 5 .

Therefore, the parameters of the quadratic function are a=1 a = 1 , b=1 b = 1 , and c=5 c = 5 .

Consequently, the correct choice from the provided options is (a=1,b=1,c=5) (a = 1, b = 1, c = 5) .

Answer

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

Exercise #13

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

Video Solution

Step-by-Step Solution

To solve the problem of identifying the coefficients in the quadratic function y=x2+x+5 y = -x^2 + x + 5 , we follow these steps:

  • Step 1: Write down the general form of a quadratic equation: y=ax2+bx+c y = ax^2 + bx + c .

  • Step 2: Compare the given equation y=x2+x+5 y = -x^2 + x + 5 to the general form.

  • Step 3: Identify the value of each coefficient:

    • The coefficient of x2 x^2 is 1-1, so a=1 a = -1 .

    • The coefficient of x x is +1+1, so b=1 b = 1 .

    • The constant term is +5+5, so c=5 c = 5 .

Therefore, the parameters of the quadratic function are a=1 a = -1 , b=1 b = 1 , and c=5 c = 5 .

This matches choice 2, confirming the parameters in the quadratic function.

Final Answer: a=1,b=1,c=5 a=-1, b=1, c=5 .

Answer

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

Exercise #14

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

Video Solution

Step-by-Step Solution

To solve this problem, we will match the given quadratic equation with its standard form:

  • Step 1: Identify the standard quadratic form as y=ax2+bx+cy = ax^2 + bx + c.
  • Step 2: Compare with given equation y=4+3x2xy = 4 + 3x^2 - x.
  • Step 3: Determine the coefficients by arranging the equation in standard form.

Let's now perform the steps:

Step 1: The standard quadratic form is y=ax2+bx+cy = ax^2 + bx + c.

Step 2: The given equation is y=4+3x2xy = 4 + 3x^2 - x.

Step 3: Rearrange the given equation to match the standard form:

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

Now, directly compare:

a=3a = 3

b=1b = -1

c=4c = 4

Therefore, the coefficients are correctly identified as a=3,b=1, a=3, b=-1, and c=4 c=4 .

The correct answer is: a=3,b=1,c=4 a=3, b=-1, c=4 .

Answer

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

Exercise #15

y=3x2+45x y=3x^2+4-5x

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Rearrange the given quadratic function into the standard form.
  • Step 2: Identify the coefficients by comparing them with the standard quadratic function.

Now let's work through each step:

Step 1: The given quadratic is y=3x2+45x y = 3x^2 + 4 - 5x . Rearrange this function to align terms with their degrees:
y=3x25x+4 y = 3x^2 - 5x + 4 .

Step 2: Compare this with the standard quadratic form y=ax2+bx+c y = ax^2 + bx + c , where:
a=3 a = 3 (the coefficient of x2x^2),
b=5 b = -5 (the coefficient of xx),
c=4 c = 4 (the constant term).

Therefore, the correct choice is a=3,b=5,c=4 a = 3, b = -5, c = 4 .

Answer

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