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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 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 about the topic. We will also solve examples based on the concept for a better understanding.
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.
The parent quadratic function is of the form f(x) = x^{2} and it connects the points whose coordinates are of the form (number, number^{2}). Transformations can be applied on this function on which it typically looks of the form f(x) = a (x  h)^{2} + k and further it can be converted into the form f(x) = ax^{2} + bx + c. Let us study each of these in detail in the upcoming sections.
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.
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 = [ b ± √(b^{2}  4ac) ] / 2a
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 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 a 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
The graph of a quadratic function is a parabola. i.e., it opens up or down in the Ushape. Here are the steps for graphing a quadratic function.
 Step  1: Find the vertex.
 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 and take two random values on either side of it.
 Step  3: Find the corresponding values of y by substituting each x value in the given quadratic function.
 Step  4: Now, we have two points on either side of the vertex so that by plotting them on the coordinate plane 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.
Example: Graph the quadratic function f(x) = 2x^{2}  8x + 3.
Solution:
By comparing this with f(x) = ax^{2} + bx + c, we get a = 2, b = 8, and c = 3.
 Step  1: Let us 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: Frame a table with vertex written in the middle row.
x y 2 5  Step  3: Fill the first column with two random numbers on either side of 2.
x y 0 1 2 5 3 4  Step  4: Find y by substituting each xvalue 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: Just plot the above points and join them by a smooth curve.
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.
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Important Notes on Quadratic Function:
 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 = [ b ± √(b^{2}  4ac) ] / 2a.
 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 zeroes of a quadratic function.
Examples of 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 vertex 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 zeros 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 zeros of quadratic function are obtained by solving f(x) = 0.
For this, we use the quadratic formula: x = [ b ± √(b^{2}  4ac) ] / 2a
x = [ 3 ± √{3^{2}  4(1)(4)}] / 2(1) = [ 3 ± √(9 + 16) ] / 2 = [ 3 ± √25 ] / 2,
x = [ 3 + 5 ] / 2, [ 3  5 ] / 2
= 1, 4
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 (multiply the binomials) 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 in Math?
A quadratic function is a polynomial function with one or more variables in which the highest exponent of the variable is two. In other words, a quadratic function is a “polynomial function of degree 2.”
Why is the Name Quadratic Function?
The meaning of "quad" is "square". Hence, a polynomial function 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 nonzero 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 the zeros of the function, the ycoordinate is 0 and the xcoordinate represents the zeros of the quadratic polynomial 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 the coordinates of the graph of the quadratic functions. We usually write the vertex of the quadratic functions in the quadratic functions in one of the rows of the table.
How to Draw Quadratic Graph?
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, 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. Finding the vertex helps in drawing a quadratic graph.
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 f(x) = 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).
What are the 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.
We can convert one of these forms into the other forms. For more information, click here.
What is the Difference Between Quadratic Function and Quadratic Equation?
A quadratic function is of the form f(x) = ax^{2} + bx + c, where a ≠ 0. Each point on its graph is of the form (x, ax^{2} + bx + c). This is for the graphing purpose. On the other hand, a quadratic equation is of the form ax^{2} + bx + c = 0, where a ≠ 0. This is for finding the solution and it gives definite values of x as solution.
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