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Parabolic Function
Parabolic function is a function of the form f(x) = ax^{2} + bx + c, and if presented in a two dimensional graphical form, it has a shape of a parabola. The equation representing a parabolic function is a quadratic equation with a second degree in x.
Let us learn more about the parabolic function, graph of a parabolic function, properties of parabolic function with the help of examples, FAQs.
1.  What Is A Parabolic Function? 
2.  Graph Of Parabolic Function 
3.  Properties Of Parabolic Function 
4.  Examples on Parabolic Function 
5.  Practice Questions 
6.  FAQs on Parabolic Function 
What Is A Parabolic Function?
Parabolic function is a function of the form f(x) = ax^{2} + bx + c. It is a quadratic expression in the second degree in x. The parabolic function has a graph similar to the parabola and hence the function is named a parabolic function.
The parabolic function has the same range value for two different domain values. The general form of a parabolic function f(x) = ax^{2} + bx + c has one f(x) value or y value for two value of x, which are x_{1}, x_{2}. The two possible points on the graph of the parabolic function is (x_{1}, y), (x_{2}, y). Hence the parabolic function can also be termed as a many to one function.
Graph Of Parabolic Function
The graph of a parabolic function is similar to a parabola. The graph of a parabola follows the basic definition of a parabola. A parabola is a locus of a point such that it is equidistant from a fixed point called the focus and the fixedline called the directrix.
The graph of a parabolic function is symmetric to a straight line, and this line is called the axis of the parabola. The axis of a parabola can be a line parallel to any of the coordinate axis or it can be a line, inclined at an angle with the coordinate axis.
Properties of Parabolic Function
The following are some of the important properties of the parabolic function, which are helpful in a better understanding of this function.
 The parabolic function has the same codomain for two different domain values.
 The set of two points that satisfy the parabolic function equation have different abscissa and the same ordinate.
 The domain of the parabolic function can be positive or negative values, but the range of the parabolic is a positive value.
 The parabolic function can also be termed as many one functions.
 The graph of a parabolic function is symmetric about a line, and this line is called the axis of the parabola.
 The equation representing the parabolic function satisfies all the properties of a geometric parabola.
Related Topics
The following topics are helpful to understand the concept of parabolic function.
Examples on Parabolic Function

Example 1: Find the parabolic function representing a parabola having the focus of (4, 0), the xaxis as the axis of the parabola, and the origin as the vertex of the parabola.
Solution:
The given focus of the parabola is (a, 0) = (4, 0)., and a = 4.
For the parabola having the xaxis as the axis and the origin as the vertex, the equation of the parabola is y^{2} = 4ax.
Hence the equation of the parabola is y^{2} = 4(4)x, or y^{2} = 16x.
And the parabolic function is y = \(4\sqrt x\)
Therefore, the equation representing the parabolic function is y = \(4\sqrt x\).

Example 2: Find the vertex of the parabolic function y = 0.5x^{2} + 3x + 4.
Solution:
Comparing the given equation with y = ax^{2} + bx + c, we have a = 0.5, b = 3, and c = 4.
The xcoordinate of the vertex is, h = b/2a = 3/2(0.5) = 3/1 = 3.
The ycoordinate of the vertex is, k = 0.5(3)^{2} + 3(3) + 4 = 0.5.
The vertex of the given parabola is, (h, k) = (3, 0.5)
Answer: The vertex of parabola = (3, 0.5).
FAQs on Parabolic Function
What Is Parabolic Function?
The parabolic function is a function of the form f(x) = ax^{2} + bx + c. It is a quadratic expression in the second degree in x. The parabolic function has a graph similar to the parabola and hence the function is named a parabolic function.
How Do You Solve A Parabolic Function?
The expression of a parabolic function is of the form f(x) = ax^{2} + bx + c, and this can be solved for x. The parabolic function is also solved similar to the quadratic function. This expression can be equalized to zero and can be either factorized or solved using the formula method.
What Is The Range Of A Parabolic Function?
The range of the parabolic function is all positive real number values. The range of a parabolic function is represented on the yaxis.
What Is The Domain Of Parabolic Function?
The domain of a parabolic function includes all the real numbers. The domain of the parabolic function is generally the xvalues of the function and are presented on both the positive and negative xaxis..
What Type Of Function Is Parabolic Function?
The parabolic function is a kind of many one functions. For a parabolic function, we have two domain values and one range value. The possible set of points on the graph of a parabolic function are (x_{1}, y), and (x_{2}, y).
What Is The Difference Between Parabolic Function And A Quadratic Function?
The parabolic function can also be considered as a quadratic function. The general expression of parabolic function is of the form f(x) = = ax^{2} + bx + c, which is similar to a quadratic expression. The quadratic expression is a second degree in both x and y, and hence every quadratic expression cannot be called a parabolic function.
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