Quadratic Function

Quadratic Functions are the type of polynomial function that has degree 2 and is a very important function as it is used in various fields of mathematical studies and also has real-life applications as well. As quadratic function is similar to parabola or we can say quadratic function are the most general parabola, thus it can be used in all the fields where parabolas and their parabolic properties can be used.

This article deals with Quadratic Functions in detail including their definition, standard form, and key features such as vertex, axis of symmetry, domain, etc. Other than these topics this will also learn to plot the graph of quadratic functions and also teach you how to shift the graph in the x and y-axis. Let’s start our journey to the world of quadratic functions.

Quadratic Function Definition

Quadratic function is the polynomial function with degree 2 and is defined as:

The polynomial function in which the highest power on the variable is 2 then, it is called a quadratic function.

Generally, the quadratic function is of the s(x) = px² + qx + r; where px² is the leading term and p is the leading coefficient of the quadratic function. q and r are some real coefficients (generally, but can be complex numbers as well), thus collectively these variables and constants are called quadratic functions.

Examples of Quadratic Function

Some examples of quadratic functions:

• f(x) = 3x² + 7x + 2
• g(x) = x² – 2
• h(x) = 9x² + 5x

General Form of Quadratic Function

The standard or general form of a quadratic function is given as follows:

f(x) = ax² + bx + c

Where,

a, b, and c are real numbers and a ≠ 0.

Learn more about the Standard form of Quadratic Equation

Vertex Form

In the vertex form of quadratic function, the quadratic function is of the form:

f(x) = a(x – h)² + k

Where a ≠ 0 and (h, k) is the vertex of the parabola that represents quadratic function.

Intercept Form

In the intercept form of quadratic function, the quadratic function is of the form:

f(x) = a (x – p) (x – q)

where, a ≠ 0 and (p, 0) and (q, 0) are the x-intercepts of the parabola representing the quadratic function.

Note: For the standard form of quadratic function i.e., f(x) = ax² + bx + c

Vertex of quadratic function = (h, k) = ((- b / 2a), f (- b / 2a))

Quadratic Function Formula

General Form of a quadratic function: f(x) = ax² + bx + c where a ≠ 0. To find the roots of the given quadratic function we apply the quadratic function formula.

x = [-b ± √ (b² – 4ac)] / 2a

Where,

• (b² – 4ac) is called the Discriminant of the quadratic function, and 
• a, b, and c are coefficients in the quadratic function.

Key Features of Quadratic Function

Some of the key features or properties of Quadratic Functions are:

• Vertex
• Axis of Symmetry
• Domain and Range
• Maximum or Minimum Value

Let’s learn about these properties in detail as follows:

Vertex

The axis of symmetry and the parabola of the quadratic function intersect at the point called the vertex of the quadratic function.

Coordinates of the vertex = (-b / a, f(-b / a))

Note: If a>0, then the direction of the parabola is upwards and if a<0, then the direction of the parabola is downwards.

Axis of Symmetry

The vertical line which passes through the vertex of the quadratic function is called the axis of symmetry. 

The axis of symmetry is defined by x = -b/2a. As the quadratic function is a general parabola, we know that the vertical line passing through the vertex of the parabola which provides symmetry to the parabola is called the axis of symmetry.

Domain and Range

The domain of the quadratic function is the values of real values of x i.e., set of all real numbers R.

The domain of the quadratic function is defined as (-∞, ∞).

The range of the quadratic function is the values of function f(x) i.e., y values. The range of the quadratic function also depends on the vertex and direction of the graph. The range of the quadratic function varies with the function. For quadratic function f(x) = ax² + bx + c, range can be defined as:

• If a > 0, then range of the quadratic function f(x) is [f(-b/2a), ∞).
• If a < 0, then range of the quadratic function f(x) is (-∞, f(-b/2a)].

Maximum or Minimum Value

The minimum or maximum value of a quadratic function is determined by the y-coordinate of the vertex. We can identify whether the value is minimum or maximum by the leading coefficient’s sign. If the leading coefficient is positive, the value is minimum and if the leading coefficient is negative, then the value is maximum. For quadratic function f(x) = ax² + bx + c, maximum or minimum value can be defined as:

If a > 0, then f(x) is minimum at x = -b/2a and the minimum value f(-b/2a) = -D/a.

If a < 0, then f(x) is maximum at x = -b/2a and the maximum value f(-b/2a) = -D/a.

Types of Quadratic Functions

There are three types of quadratic functions:

• Univariate quadratic functions
• Bivariate quadratic functions
• Multivariate quadratic functions

Univariate Quadratic Functions

The quadratic function that involves only one variable is called the univariate quadratic function and all the discussion throughout this article involves only this type of quadratic function. Thus, the general form of univariate quadratic function is given as:

f(x) = px² + qx + r

Where,

• x is the variables, and 
• p, q, and r are the coefficients of variables.

Bivariate Quadratic Function

The quadratic function that involves two variables is called the bivariate quadratic function and the general form of  bivariate quadratic function is given as:

f(x) = ax² + by² + cx + dy + exy + f

Where,

• x, and y are variables, and 
• a, b, c, d, e, and f are the coefficients of variables.

Multivariate Quadratic Function

The quadratic function that involves three or more variables is called the multivariate quadratic function. The general form of a multivariate quadratic function with three variables is given as:

f(x) = ax² + by² + cz² + dx + ey + fxy + gyz + hxz + i

Where,

• x, y, and z are variables, and 
• a, b, c, d, e, f, g, h, and i are the coefficients of variables.

Graphing Quadratic Function

The graph of the quadratic function is a U-shaped parabola whose direction is either upwards or downwards.

Steps to plot a graph of a quadratic function:

1. Find the vertex of the quadratic function.
2. Construct the table for different values of x and substitute it to find the value of quadratic function f(x) i.e., y.
3. Plot the points in the graph and join them to get a graph for the given quadratic function.

Example: Plot the graph for the quadratic function f(x) = x² – x – 6.

Solution:

For function, f(x) = x² – x – 6

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

Step 1: The vertex of above quadratic function = (-b / a, f(-b/a))

f(-b/a) = f [-(-1)/1] = f(1) = -6

The vertex of above quadratic function = (1, -6)

Step 2: Following is the quadratic function table

x	-2	-1	0	1	2
y	0	– 4	-6	-6	-4

Step 3: Plot the graph from above table.

Shifting of Graph

By changing the vertex we can shift the graph in the cartesian plane anywhere, we can make two kinds of shifts by changing the vertex parameters i.e., 

• Horizontal shift
• Vertical shift

Horizontal Shift

The quadratic function graph of f(x) = (x – h)² shifts the graph of f(x) = x² by h units horizontally.

• If h>0, then shift the parabola h units towards the left.
• If h<0, then shift the parabola h units towards the right.

Vertical Shift

The quadratic function graph of f(x) = x² + k shifts the graph of f(x) = x² by k units vertically.

• If k>0, then shift the parabola k units upwards.
• If k<0, then shift the parabola |k| units downwards.

Solving Quadratic Equations

For a quadratic function, f(x) we can form it into any quadratic equation by equating it to any quadratic, linear of constant function i.e., f(x) = g(x) is a quadratic equation if g(x) is a function with at most degree 2. To solve such an equation, we have various methods such as:

• Factorization Method
• Completing Square Method
• Quadratic Formula Method

Learn the details of the solution by reading quadratic equations

Real and Complex Solutions

In quadratic equation ax² + bx + c where, a ≠ 0, the discriminant of the equation is given by:

Discriminant (D) = b² – 4ac

The nature of the roots depends on the discriminant of the quadratic equation.

• If D > 0 then, the roots are real and distinct.
• If D = 0 then, the roots are real and equal.
• If D < 0 then, there are no real roots of the given equation or only complex or imaginary roots exist.

Solved Examples on Quadratic Function

Problem 1: Find the vertex of the quadratic function f(x) = 5(x – 3)² + 6

Solution:

f(x) = 5(x – 3)² + 6

The above quadratic function represents the vertex form of the quadratic equation which can be written as:

f(x) = a(x – h)² + k

where, (h, k) is the vertex of quadratic function.

Here, h = 3 and k = 6

The vertex of the quadratic equation f(x) = 5(x – 3)² + 6 is (3, 6).

Problem 2: Find the roots of the quadratic function f(x) = x² + 5x + 6 using the quadratic function formula.

Solution:

f(x) = x² + 5x + 6

The quadratic equation formula: x = [-b ± √ (b² – 4ac)] / 2a

Here, a =1, b =5 and c= 6

x = [-5 ± √ (5² – 4 × 1 × 6)] / 2 × 1

x = [-5 ± √ (25 – 24)] / 2

x = [-5 ± √1)] / 2

x = [-5 ± 1)] / 2

x = [-5 + 1)] / 2 or x = [-5 – 1)] / 2

x = -4 / 2 or x = -6 / 2

x = -2 or x = -3

Problem 3: Convert the quadratic function f(x) = (x – 4) (x + 5) in standard form.

Solution:

f(x) = (x – 4) (x + 5)

Multiplying both brackets

f(x) = x(x + 5) – 4(x + 5)

f(x) = x² + 5x – 4x – 20

f(x) = x² + x – 20

The quadratic function f(x) = (x – 4) (x + 5) in standard form is f(x) = x² + x – 20

Problem 4: Solve: f(x) = x² + 4x – 45 using quadratic formula.

Solution:

f(x) = x² + 4x – 45

The quadratic equation formula: x = [-b ± √ (b² – 4ac)] / 2a

Here, a =1, b = 4 and c = -45

x = [-4 ± √ (4² – 4 × 1 × -45)] / 2 × 1

x = [-4 ± √ (16 + 180)] / 2

x = [-4 ± √196)] / 2

x = [-4 ± 14)] / 2

x = [-4 + 14)] / 2 or x = [-4 – 14)] / 2

x = -10 / 2 or x = -18 / 2

x = -5 or x = -9

FAQs on Quadratic Function

Q1: What are Quadratic Functions?

Answer:

Quadratic functions are the polynomial function with degree 2.

Q2: Write the Standard Form of the Quadratic Function.

Answer:

The standard form of quadratic equation is: f(x) = ax² + bx + c where, a, b and c are real numbers and a ≠ 0

Q3: Write the Formula for Calculating the Roots of the Quadratic Function.

Answer:

The formula for calculating the roots of the quadratic function (f(x) = ax² + bx + c) is given by:

x = [-b ± √ (b² – 4ac)] / 2a

Q4: What are the Different forms of Quadratic Equations?

Answer:

Different forms of quadratic equation are:

• Standard form
• Vertex form
• Intercept form

Q5: What is the Direction of the Parabola if a>0 and a<0, where a is the Leading Coefficient of the Quadratic Function?

Answer:

The direction of the parabola is upwards (opens upwards) when a > 0 and open downwards when a < 0.