The Quadratic Function - Examples, Exercises and Solutions

Understanding The Quadratic Function

Complete explanation with examples

The Parabola y=ax2+bx+c y=ax^2+bx+c 

This function is a quadratic function and is called a parabola.

We will focus on two main types of parabolas: maximum and minimum parabolas.

Minimum Parabola

Also called smiling or happy.

A vertex is the minimum point of the function, where YY is the lowest.

We can identify that it is a minimum parabola if the aa equation is positive.

1b - We can identify that it is a minimum parabola if the equation a is positive


Maximum Parabola

Also called sad or crying.

A vertex is the maximum point of the function, where YY is the highest.

We can identify that it is a maximum parabola if the aa equation is negative.

2b - We can identify that it is a maximum parabola if the a equation is negative

To the parabola,

the vertex marks its highest point.

How do we find it?


Detailed explanation

Practice The Quadratic Function

Test your knowledge with 17 quizzes

\( y=-4x^2+3 \)

Examples with solutions for The Quadratic Function

Step-by-step solutions included
Exercise #1

y=x2+10x 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 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

Video Solution
Exercise #2

y=2x25x+6 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.

Answer:

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

Video Solution
Exercise #3

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

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

Video Solution
Exercise #4

y=5x24x30 y=5x^2-4x-30

Step-by-Step Solution

The given quadratic function is y=5x24x30 y = 5x^2 - 4x - 30 .
To identify the coefficients, let's compare this with the standard form of a quadratic equation, y=ax2+bx+c y = ax^2 + bx + c .

  • Step 1: Compare the terms of the given equation to the standard form.
  • Step 2: Identify each coefficient:
    For ax2 ax^2 , a=5 a = 5 in the term 5x2 5x^2 .
    For bx bx , b=4 b = -4 in the term 4x-4x.
    For the constant term c c , c=30 c = -30 .

Thus, we have identified the coefficients as a=5 a = 5 , b=4 b = -4 , and c=30 c = -30 .

Therefore, the correct answer is a=5,b=4,c=30 a = 5, b = -4, c = -30 .

The correct choice is .

Answer:

a=5,b=4,c=30 a=5,b=-4,c=-30

Video Solution
Exercise #5

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

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

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

Video Solution

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