What Is a Quadratic Function? Definition, Formula, Graph, Domain and Range of Quadratic Function

What Is a Quadratic Function?

A quadratic function is one of the most important topics in algebra. It is widely used in mathematics, physics, engineering, economics, and computer science. Students study quadratic functions to understand curved graphs, maximum and minimum values, and many real-life situations involving motion and optimization.

A quadratic function is a function whose highest power of the variable is 2. Unlike a linear function, which produces a straight-line graph, the graph of a quadratic function is always a curved shape called a parabola.

Understanding the quadratic function helps students solve mathematical problems involving equations, graphing, and practical applications.

Define Quadratic Function

To define quadratic function, we use the following standard form:$$f(x)=ax^2+bx+c$$f(x)=ax2+bx+c

Where

• $a$a, $b$b, and $c$c are real numbers.
• $a \neq 0$a=0.
• $x$x is the independent variable.
• $f(x)$f(x) is the dependent variable.

The condition $a \neq 0$a=0 is necessary because if $a=0$a=0, the equation becomes linear instead of quadratic.

Examples

Example 1$$f(x)=x^2+5x+6$$f(x)=x2+5x+6

This is a quadratic function because the highest power of $x$x is 2.

Example 2$$f(x)=3x^2-7x+4$$f(x)=3×2−7x+4

This is also a quadratic function.

Example 3$$f(x)=-2x^2+8$$f(x)=−2×2+8

This is a quadratic function since the coefficient of $x^2$x2 is not zero.

Standard Form of a Quadratic Function

The standard form is$$f(x)=ax^2+bx+c$$f(x)=ax2+bx+c

Where

• a is the coefficient of $x^2$x2.
• b is the coefficient of $x$x.
• c is the constant term.

Example$$f(x)=2x^2+3x-5$$f(x)=2×2+3x−5

Here

• $a=2$a=2
• $b=3$b=3
• $c=-5$c=−5

Characteristics of a Quadratic Function

The following are the main characteristics of a quadratic function.

• Highest exponent is 2.
• Graph is always a parabola.
• May open upward or downward.
• Has one vertex.
• Has one axis of symmetry.
• Can have zero, one, or two real roots.

Graphing Quadratic Function

Graphing quadratic function means drawing the parabola on the coordinate plane.

The shape of the graph depends mainly on the value of $a$a.

Case 1: Positive Value of a

If$$a>0$$a>0

the parabola opens upward.

Example$$f(x)=x^2$$f(x)=x2

The graph has a minimum point.

Case 2: Negative Value of a

If$$a<0$$a<0

the parabola opens downward.

Example$$f(x)=-x^2$$f(x)=−x2

The graph has a maximum point.

Steps for Graphing Quadratic Function

1. Write the function.
2. Find the vertex.
3. Find the axis of symmetry.
4. Calculate several points.
5. Plot the points.
6. Draw a smooth parabola through the points.

Vertex of a Quadratic Function

The vertex is the turning point of the parabola.

Its x-coordinate is$$x=\frac{-b}{2a}$$x=2a−b​

After finding the x-coordinate, substitute it into the function to obtain the y-coordinate.

Example$$f(x)=x^2-6x+5$$f(x)=x2−6x+5

Here$$a=1,\quad b=-6$$a=1,b=−6

So,$$x=\frac{-(-6)}{2(1)}=3$$x=2(1)−(−6)​=3

Now,$$f(3)=9-18+5=-4$$f(3)=9−18+5=−4

Therefore, the vertex is$$(3,-4)$$(3,−4)

Axis of Symmetry

The axis of symmetry divides the parabola into two equal parts.

Its equation is$$x=\frac{-b}{2a}$$x=2a−b​

Domain and Range of Quadratic Function

Domain of Quadratic Function

The domain and range of quadratic function are important concepts.

The domain of every quadratic function is$$(-\infty,\infty)$$(−∞,∞)

This means every real number can be substituted for $x$x.

Range of Quadratic Function

The range depends on the direction of the parabola.

If the parabola opens upward,$$y\geq \text{minimum value}$$y≥minimum value

If the parabola opens downward,$$y\leq \text{maximum value}$$y≤maximum value

Example$$f(x)=x^2$$f(x)=x2

Domain$$(-\infty,\infty)$$(−∞,∞)

Range$$[0,\infty)$$[0,∞)

Real-Life Applications of Quadratic Functions

Quadratic functions are used in many fields.

• Projectile motion
• Bridge construction
• Satellite dishes
• Economics
• Business profit analysis
• Engineering
• Architecture
• Sports science
• Computer graphics

Difference Between Linear and Quadratic Function

Linear Function	Quadratic Function
Highest power is 1	Highest power is 2
Graph is a straight line	Graph is a parabola
Constant rate of change	Variable rate of change
Standard form: $ax+b$ax+b	Standard form: $ax^2+bx+c$ax2+bx+c

Importance of Learning Quadratic Functions

Quadratic functions form the foundation for higher mathematics. Students studying algebra, calculus, engineering, statistics, economics, and computer science frequently use quadratic functions. A strong understanding of this topic makes solving equations and graphing much easier.

Students can also explore more mathematics concepts through Nisar Math Academy, where recorded lectures, lesson plans, assessments, and downloadable notes are available. Free Class 9 Mathematics notes for the first exercise of each chapter and selected free lectures are provided. Students who want complete access to all recorded lectures, notes, assessments, and premium learning material can join the website membership.

Conclusion

A quadratic function is a polynomial function of degree two written as $f(x)=ax^2+bx+c$f(x)=ax2+bx+c, where $a\neq0$a=0. Its graph is a parabola that may open upward or downward depending on the sign of $a$a. Learning how to define quadratic function, perform graphing quadratic function, and determine the domain and range of quadratic function helps students solve algebraic problems and understand many real-world applications.

Short Notes

Quadratic Function Short Notes

• A quadratic function has degree 2.
• Standard form is $f(x)=ax^2+bx+c$f(x)=ax2+bx+c.
• Here, $a\neq0$a=0.
• Its graph is called a parabola.
• If $a>0$a>0, the parabola opens upward.
• If $a<0$a<0, the parabola opens downward.
• Vertex is the turning point.
• Axis of symmetry is $x=\frac{-b}{2a}$x=2a−b​.
• Domain is all real numbers.
• Range depends on the vertex and direction of the parabola.

MCQs

1. A quadratic function has highest power equal to

A) 1

B) 2

C) 3

D) 4

Answer: B

2. Standard form of a quadratic function is

A) $ax+b$ax+b

B) $ax^2+bx+c$ax2+bx+c

C) $a+x$a+x

D) $a^2+b^2$a2+b2

Answer: B

3. The graph of a quadratic function is called

A) Circle

B) Ellipse

C) Parabola

D) Triangle

Answer: C

4. If a>0a>0a>0, the parabola opens

A) Downward

B) Upward

C) Left

D) Right

Answer: B

5. Domain of a quadratic function is

A) Positive numbers

B) Negative numbers

C) All real numbers

D) Integers only

Answer: C

6. The axis of symmetry is

A) $x=\frac{-b}{2a}$x=2a−b​

B) $x=a+b$x=a+b

C) $y=ax$y=ax

D) $y=x$y=x

Answer: A

7. Which is a quadratic function?

A) $2x+5$2x+5

B) $x^3+1$x3+1

C) $4x^2-3x+7$4×2−3x+7

D) $5$5

Answer: C

8. The turning point of a parabola is called

A) Root

B) Vertex

C) Intercept

D) Origin

Answer: B

9. If a<0a<0a<0, the parabola opens

A) Upward

B) Downward

C) Left

D) Right

Answer: B

10. Which coefficient must not be zero?

A) $b$b

B) $c$c

C) $a$a

D) None

Answer: C

Worksheet / Assignment

Part A: Fill in the Blanks

1. A quadratic function has degree ______.
2. The graph of a quadratic function is called ______.
3. The standard form is ______.
4. The domain of every quadratic function is ______.
5. The turning point is called the ______.

Part B: Short Questions

1. Define quadratic function.
2. Write the standard form of a quadratic function.
3. What is a parabola?
4. What is the domain of a quadratic function?
5. What is the range of $f(x)=x^2$f(x)=x2?
6. Write the formula of the axis of symmetry.
7. What happens when $a<0$a<0?
8. State two real-life applications of quadratic functions.

Part C: Solve the Following

1. Identify whether the following are quadratic functions.
   • $x^2+4x+1$x2+4x+1
   • $5x+3$5x+3
   • $2x^2-9$2×2−9
   • $x^3+2$x3+2
2. Find the vertex of
   • $f(x)=x^2-4x+3$f(x)=x2−4x+3
3. Find the axis of symmetry of
   • $f(x)=2x^2+8x+5$f(x)=2×2+8x+5
4. State the domain and range of
   • $f(x)=x^2+1$f(x)=x2+1

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