Quadratic Function
Quadratic functions are used in different fields of engineering and science to obtain values of different parameters. Graphically, they are represented by a parabola. Depending on the coefficient of the highest degree, the direction of the curve is decided. The word "Quadratic" is derived from the word "Quad" which means square. In other words, a quadratic function is a “ polynomial function of degree 2.” There are many scenarios where quadratic functions are used.
Did you know that when a rocket is launched, its path is described by the solution of a quadratic function? In this article, we will explore the world of quadratic functions in math. You will get to learn about the graphs of quadratic functions, quadratic functions formula, and other interesting facts around the topic.
What is Quadratic Function?
A quadratic function is a polynomial function with one or more variables in which the highest power of the variable is two. Since the highest degree term in a quadratic function is of the second degree, therefore it is also called the polynomial of degree 2. A quadratic function has a minimum of one term which is of the second degree.
Quadratic Function Definition
A quadratic function is of the form f(x) = ax^{2} + bx + c, where a, b, and c are real numbers with a ≠ 0. Let us see a few examples of quadratic functions:
 f(x) = 2x^{2} + 4x  5; Here a = 2, b = 4, c = 5
 f(x) = 3x^{2}  9; Here a = 3, b = 0, c = 9
 f(x) = x^{2}  x; Here a = 1, b = 1, c = 0
Now, consider f(x) = 4x11; Here a = 0, therefore f(x) is not a quadratic function.
Standard Form of Quadratic Function
The standard form of a quadratic function is f(x)=a(xh)^{2}+k, where (h,k) is the vertex of the parabola. The value of 'a' determines the direction of the graph of the parabola. Sometimes, a quadratic function is not written in its standard form. It is written as f(x) = ax^{2} + bx + c and we may have to change it into the standard form.
The parabola opens upwards or downwards as per the value of 'a' varies:
 If a>0, then the parabola opens upward.
 If a<0, then the parabola opens downward.
Specific Forms of Quadratic Function
There are three main forms of quadratic functions. Let us see each form and its expression:
 General Form: f(x) = ax^{2} + bx + c, where a, b, c are real numbers.
 Factored Form: f(x) = (ax+b) (cx+d), where a, b, c, d are real numbers
 Vertex Form: f(x) = a(x  h)^{2} + k, where a, h, k are real numbers. This form is also called the standard form of quadratic function.
Domain and Range of Quadratic Function
A quadratic function is a polynomial function that is defined for all real numbers. So, the domain of a quadratic function is the set of real numbers, that is, R. The range of the quadratic function depends on the graph's opening side and vertex. So, look for the lowermost and uppermost f(x) values on the graph of the function to determine the range of the quadratic function. Hence, the range of quadratic functions is the set of all real values that we can get by plugging real numbers into x.
Graphing Quadratic Function
Now, in terms of graphing quadratic functions, we will understand a stepbystep procedure to plot the graph of any quadratic function. Consider the general quadratic function f(x) = ax^{2} + bx + c. First, we rearrange it (by the method of completion of squares) to the following form: f(x) = a(x + b/2a)^{2}  D/4a. The term D is the discriminant, given by D = b^{2}  4ac. Now, to plot the graph of f(x), we start by taking the graph of x^{2},and applying a series of transformations to it:
 Step 1: x^{2} to ax^{2}: This will imply a vertical scaling of the original parabola. If a is negative, the parabola will also flip its mouth from the positive to the negative side. The magnitude of the scaling depends upon the magnitude of a.
 Step 2: ax^{2} to a(x + b/2a)^{2}: This is a horizontal shift of magnitude b/2a units. The direction of the shift will be decided by the sign of b/2a. The new vertex of the parabola will be at (b/2a,0). The following figure shows an example shift:
 Step 3: a(x + b/2a)^{2} to a(x + b/2a)^{2}  D/4a: This transformation is a vertical shift of magnitude D/4a units. The direction of the shift will be decided by the sign of D/4a. The final vertex of the parabola will be at (b/2a, D/4a). The following figure shows an example shift:
The graph of quadratic functions can also be obtained using the quadratic functions calculator.
Maxima and Minima of Quadratic Function
Maxima and minima of quadratic functions can be found by using differentiation. To understand the concept better, let us consider an example and solve it. Let's take an example of quadratic function f(x) = 3x^{2} + 4x + 7.
Differentiating the function,
⇒f'(x) = 6x + 4
Equating it to zero,
⇒6x + 4 = 0
⇒Therefore, x = 2/3
Double differentiating the function,
⇒f''(x) = 6 > 0
Since the double derivative of the function is greater than zero, we will have minima at x = 2/3, and the parabola is upwards.
Similarly, if the double derivative at the stationary point is less than zero, then the function would have maxima. Hence, by using differentiation, we can find the minimum or maximum of a quadratic function.
Quadratic Functions Formula
A quadratic function can always be factorized, but the factorization process may be difficult if the zeroes of the expression are noninteger real numbers, or nonreal numbers. In such cases, we can use the quadratic formula to determine the zeroes of the expression. The general form of a quadratic function is given as: f(x) = ax^{2} + bx + c, where a, b, and c are real numbers with a ≠ 0. The roots of the quadratic function f(x) can be calculated using the formula of the quadratic function which is:
\(x=\dfrac{b \pm \sqrt{b^24ac}}{2a}\)
Related Topics
 Quadratic Equations Calculator
 Factorization of Quadratic Equations
 Roots of Quadratic Equation Calculator
Important Notes:
 The standard form of the quadratic function is f(x)=ax^{2}+bx+c where a≠0.
 The curve of the quadratic function is in the form of a parabola.
 The quadratic formula is given by \(\begin{align} \frac{b \pm \sqrt{b^24ac}}{2a}\end{align}\).
 The discriminant is given by b^{2}4ac. This is used to determine the nature of the solutions of a quadratic function.
Examples on Quadratic Function

Example 1: Determine the vertex of the quadratic function f(x) = 2(x+3)^{2}  2.
Solution: We have f(x) = 2(x+3)^{2}  2 which can be written as f(x) = 2(x(3))^{2} + (2)
Comparing the given quadratic function with the standard form of quadratic function f(x) = a(xh)^{2} + k, where (h,k) is the vertex of the parabola, we have
h = 3, k = 2
Hence, the vertex of f(x) is (3,2)
Answer: Vertex = (3,2)

Example 2: Solve the quadratic function f(x) = x^{2} + 3x  4 using the quadratic functions formula.
Solution: The quadratic function f(x) = x^{2} + 3x  4. On comparing f(x) with the general form ax^{2} + bx + c, we get a = 1, b = 3, c = 4
Substituting the values in the quadratic functions formula to solve f(x) = 0 is \(x=\dfrac{b \pm \sqrt{b^24ac}}{2a}\)
\(\begin{align} x &= \dfrac{3 \pm \sqrt{3^24\times 1\times (4)}}{2} \\&= \dfrac{3\pm\sqrt{25}}{2}\\&=\dfrac{35}{2}, \dfrac{3+5}{2}\\&=4, 1 \end{align} \)
Answer: Roots of f(x) = x^{2} + 3x  4 are 1 and 4

Example 3: Write the quadratic function f(x) = (x12)(x+3) in the general form ax^{2} + bx + c.
Solution: We have the quadratic function f(x) = (x12)(x+3). We will solve the function to wrtie it in the general form.
f(x) = (x12)(x+3)
= x(x+3)  12(x+3)
= x^{2 }+ 3x  12x  36
= x^{2}  9x  36
Here a = 1, b = 9, c = 36
Answer: x^{2}  9x  36
FAQs on Quadratic Function
What Is the Vertex of a Quadratic Function?
Vertex of a quadratic function is a point where the parabola changes direction and crosses the axis of symmetry. It is a point where the parabola changes from maxima to minima. At this point, the derivative of the quadratic functions changes sign.
What Are the Zeros of a Quadratic Function?
The zeroes of a quadratic function are points where the graph of the function intersects the Xaxis. At zeros of the function, the ycoordinate is 0 and the xcoordinate represents the zeros of the quadratic function. The zeros of a quadratic function are also called the roots of the function.
What is a Quadratic Functions Table?
A quadratic functions table is a table where we determine the values of ycoordinates corresponding to each xcoordinates and viceversa. The table consists of coordinates of the graph of the quadratic functions. We find the vertex of the quadratic functions using the quadratic functions table.
How to Graph Quadratic Functions?
The graph of a quadratic function is a parabola. It can be drawn by plotting the coordinates on the graph. We plug in the values of x and obtain the corresponding values of y, and hence obtaining the coordinates of the graph. After plotting the coordinates on the graph, we connect the dots using a free hand to obtain the graph of the quadratic functions.
How to Find the xintercept of a Quadratic Function?
The Xintercept of a quadratic function can be found considering the quadratic function f(x) = 0 and then determining the value of x. In other words, the xintercept is nothing but zeroes of a quadratic equation.
Is Parabola is a Quadratic Function?
A parabola is a graph of a quadratic function. A quadratic function is of the form ax^{2} + bx + c with a not equal to 0. Parabola is a Ushaped or inverted Ushaped graph of a quadratic function.
How to Find the Inverse of a Quadratic Function?
The inverse of a quadratic function f(x) can be found by replacing f(x) by y. Then, we switch the roles of x and y, that is, we replace x with y and y with x. After this, we solve y for x and then replace y by f^{1}(x) to obtain the inverse of the quadratic function f(x).