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 formulas, 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 exponent 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. It is an algebraic function.
Standard Form of a Quadratic Function
The standard form of a quadratic function is of the form f(x) = ax^{2} + bx + c, where a, b, and c are real numbers with a ≠ 0.
Quadratic Function examples
The quadratic function equation is f(x) = ax^{2} + bx + c, where 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.
Vertex of Quadratic Function
The vertex of a quadratic function (which is in U shape) is where the function has a maximum value or a minimum value. The axis of symmetry of the quadratic function intersects the function (parabola) at the vertex.
Different Forms of Quadratic Function
A quadratic function can be in different forms: standard form, vertex form, and intercept form. Here are the general forms of each of them:
 Standard form: f(x) = ax^{2} + bx + c, where a ≠ 0.
 Vertex form: f(x) = a(x  h)^{2} + k, where a ≠ 0 and (h, k) is the vertex of the parabola representing the quadratic function.
 Intercept form: f(x) = a(x  p)(x  q), where a ≠ 0 and (p, 0) and (q, 0) are the xintercepts of the parabola representing the quadratic function.
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.
We can always convert one form to the other form. We can easily convert vertex form or intercept form into standard form by just simplifying the algebraic expressions. Let us see how to convert the standard form into each of vertex form and intercept form.
Converting Standard Form of Quadratic Function Into Vertex Form
A quadratic function f(x) = ax^{2} + bx + c can be easily converted into the vertex form f(x) = a (x  h)^{2} + k by using the values h = b/2a and k = f(b/2a). Here is an example.
Example: Convert the quadratic function f(x) = 2x^{2}  8x + 3 into the vertex form.
 Step  1: By comparing the given function with f(x) = ax^{2} + bx + c, we get a = 2, b = 8, and c = 3.
 Step  2: Find 'h' using the formula: h = b/2a = (8)/2(2) = 2.
 Step  3: Find 'k' using the formula: k = f(b/2a) = f(2) = 2(2)^{2}  8(2) + 3 = 8  16 + 3 = 5.
 Step  4: Substitute the values into the vertex form: f(x) = 2 (x  2)^{2}  5.
Converting Standard Form of Quadratic Function Into Intercept Form
A quadratic function f(x) = ax^{2} + bx + c can be easily converted into the vertex form f(x) = a (x  p)(x  q) by using the values of p and q (xintercepts) by solving the quadratic equation ax^{2} + bx + c = 0.
Example: Convert the quadratic function f(x) = x^{2}  5x + 6 into the intercept form.
 Step  1: By comparing the given function with f(x) = ax^{2} + bx + c, we get a = 1.
 Step  2: Solve the quadratic equation: x^{2}  5x + 6 = 0
By factoring the left side part, we get
(x  3) (x  2) = 0
x = 3, x = 2  Step  3: Substitute the values into the intercept form: f(x) = 1 (x  3)(x  2).
Domain and Range of Quadratic Function
The domain of quadratic function is the set of all xvalues that makes the function defined and the range of a quadratic function is the set of all yvalues that the function results in by substituting different xvalues.
Domain of Quadratic Function
A quadratic function is a polynomial function that is defined for all real values of x. So, the domain of a quadratic function is the set of real numbers, that is, R. In interval notation, the domain of any quadratic function is (∞, ∞).
Range of Quadratic Function
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. The range of any quadratic function with vertex (h, k) and the equation f(x) = a(x  h)^{2} + k is:
 y ≥ k (or) [k, ∞) when a > 0 (as the parabola opens up when a > 0).
 y ≤ k (or) (∞, k] when a < 0 (as the parabola opens down when a < 0).
Graphing Quadratic Function
Now, in terms of graphing quadratic functions, we will understand a stepbystep procedure to plot the graph of any quadratic function. The steps are explained through an example where we are going to graph the quadratic function f(x) = 2x^{2}  8x + 3. By comparing this with f(x) = ax^{2} + bx + c, we get a = 2, b = 8, and c = 3.
 Step  1: Find the vertex.
xccordinate of vertex = b/2a = 8/4 = 2
ycoordinate of vertex = f(b/2a) = 2(2)^{2}  8(2) + 3 = 8  16 + 3 = 5.
Therefore, vertex = (2, 5).  Step  2: Compute a quadratic function table with two columns x and y with 5 rows (we can take more rows as well) with vertex to be one of the points as follows:
x y 2 5  Step  3: Take any two random numbers for x on the left side of the xcoordinate of vertex and two random numbers on the right side of the xcoordinate of vertex as follows:
x y 0 1 2 5 3 4  Step  4: Find the corresponding values of y by substituting each x value in the given quadratic function. For example, when x = 0, y = 2(0)^{2}  8(0) + 3 = 3.
x y 0 3 1 3 2 5 3 3 4 3  Step  5: Now, we have two points on either side of the vertex so that by plotting them and joining them by a curve, we can get the perfect shape. Also, extend the graph on both sides. Here is the quadratic function graph.
Note: We can plot the xintercepts and yintercept of the quadratic function as well to get a neater shape of the graph.
The graph of quadratic functions can also be obtained using the quadratic functions calculator.
Maxima and Minima of Quadratic Function
Maxima or minima of quadratic functions occur at its vertex. It can also 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
⇒ 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 (by second derivative test), 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 graph of the quadratic function is in the form of a parabola.
 The quadratic formula is used to solve a quadratic equation ax^{2} + bx + c = 0 and is given by x = \(\begin{align} \frac{b \pm \sqrt{b^24ac}}{2a}\end{align}\).
 The discriminant of a quadratic equation ax^{2} + bx + c = 0 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: Find the roots of 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
The roots of quadratic function are obtained by solving f(x) = 0.
For this, we use the quadratic formula: \(x=\dfrac{b \pm \sqrt{b^24ac}}{2a}\)
\(\begin{align} x &= \dfrac{3 \pm \sqrt{3^24\times 1\times (4)}}{2} \\&= \dfrac{3\pm\sqrt{25}}{2}\\&=\dfrac{35}{2}, \dfrac{3+5}{2}\\&=4, 1 \end{align} \)
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 just expand it to write 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
Answer: x^{2}  9x  36
FAQs on Quadratic Function
What is Quadratic Function Definition?
The meaning of "quad" means "square. Hence, a polynomial of degree 2 is called a quadratic function.
What is Quadratic Function Equation?
A quadratic function is a polynomial of degree 2 and so the equation of quadratic function is of the form f(x) = ax^{2} + bx + c, where 'a' is a non zero number; and a, b, and c are real numbers.
What Is the Vertex of 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 increasing to decreasing or from decreasing to increasing. At this point, the derivative of the quadratic function is 0.
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 usually write the vertex of the quadratic functions in 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 zero 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).
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