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.

1. What is Quadratic Function?
2. Standard Form of a Quadratic Function
3. Quadratic Functions Formula
4. Different Forms of Quadratic Function
5. Domain and Range of Quadratic Function
6. Graphing Quadratic Function
7. Maxima and Minima of Quadratic Function
8. FAQs on Quadratic Function

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

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:

Now, consider f(x) = 4x-11; 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 function 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 non-integer real numbers or non-real 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:

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:

The parabola opens upwards or downwards as per the value of 'a' varies:

Shape of quadratic function

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.

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 (x-intercepts) 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.

Domain and Range of Quadratic Function

The domain of a quadratic function is the set of all x-values that makes the function defined and the range of a quadratic function is the set of all y-values that the function results in by substituting different x-values.

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:

Graphing Quadratic Function

The graph of a quadratic function is a parabola. i.e., it opens up or down in the U-shape. Here are the steps for graphing a quadratic function.

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.

x y
2 -5
x y
0
1
2 -5
3
4
x y
0 3
1 -3
2 -5
3 -3
4 3

Quadratic function graph

  • Step - 5: Just plot the above points and join them by a smooth curve.
  • Note: We can plot the x-intercepts and y-intercept 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,

    Equating it to zero,

    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:

    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(x-h) 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 ± √] / 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) = (x-12)(x+3) in the general form ax 2 + bx + c. Solution: We have the quadratic function f(x) = (x-12)(x+3). We will just expand (multiply the binomials) it to write it in the general form. f(x) = (x-12)(x+3) = x(x+3) - 12(x+3) = x 2 + 3x - 12x - 36 = x 2 - 9x - 36 Answer: x 2 - 9x - 36

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