## Quadratic function

A second degree polynomial function is called Quadratic function. The general form of a quadratic function is

f(x) = ax^{2} + bx + c, a ≠ 0.

Example: -

f(x) = 3x^{2} + 4x + 5 is an example of a quadratic function

f(x) = 6x^{2} + 7x +12is another example of quadratic function

f(x) = a(x - h)

^{2}+ k, where a ≠ 0.

This form is also known as vertex form of a quadratic function.

Example: -

f(x) = 3(x-2)

^{2}+ 7 is in the standard form.

If f(x) = a(x - h)

^{2}+ k, is the given quadratic function, then the vertex of the function is (h, k).

Usually, we are given the general form of a quadratic function. To convert the quadratic function in the general form to standard form, we follow the steps below: -

- Factor out the coefficient a from the first two terms in the quadratic function f(x) = ax
^{2}+ bx + c - Complete the square for the terms in the parenthesis.
- Simplify

Lets consider a few examples.

Example 1: -

Convert the quadratic function f(x) = x^{2} + 7x + 12 in the standard form.

Solution: -

Given quadratic function is f(x) = x^{2} + 8x + 12.

Comparing with the general form of the quadratic function, f(x) = ax^{2} + bx + c, we get

a = 1.

Since a = 1, we complete the square of the first two terms in the quadratic equation. To complete the square, we add and subtract the square of the half of the second term.

So we get

f(x) = x^{2} + 8x + 16 - 16 + 12

Now we can combine the first three terms and last two term.

We get,**f(x) = (x + 4) ^{2} - 4, which is the required standard form.**

Example 2: -

Convert the quadratic function f(x) = 3x

^{2}+ 6x + 12 in the standard form.

Solution: -

Given quadratic function is f(x) = 3x

^{2}+ 6x + 12.

Comparing with the general form of the quadratic function, f(x) = ax

^{2}+ bx + c, we get

a = 3.

Since a = 3, we first take out 3 common from the first two terms

We get, f(x) = 3(x

^{2}+ 2x )+ 12.

Now we complete the square of the first two terms in the quadratic equation. To complete the square, we add and subtract the square of the half of the second term.

So we get

f(x) = 3(x

^{2}+ 2x + 1 - 1) + 12

=3(x

^{2}+ 2x + 1) - 3 + 12

Now we can combine the first three terms and last two term.

We get,

**f(x) = 3(x + 1)**

^{2}+ 9, which is the required standard form.Example 3: -

Convert the quadratic function f(x) = 5x

^{2}- 15x + 18 in the standard form.

Solution: -

Given quadratic function is f(x) = 5x

^{2}- 15x + 18.

Comparing with the general form of the quadratic function, f(x) = ax

^{2}+ bx + c, we get

a = 5.

Since a = 5, we first take out 5 common from the first two terms

We get, f(x) = 5(x

^{2}- 3x )+ 18.

Now we complete the square of the first two terms in the quadratic equation. To complete the square, we add and subtract the square of the half of the second term.

So we get

f(x) = 5(x

^{2}- 3x + 9/4 - 9/4) + 12

= 5(x

^{2}- 3x + 9/4) - 45/4 + 12

Now we can combine the first three terms and last two term.

We get,

**f(x) = 5(x - 3/2)**

^{2}+ 3/4, which is the required standard form.Example 4: -

Convert the quadratic function f(x) = 4x

^{2}- 24x + 18 in the standard form.

Solution: -

Given quadratic function is f(x) = 4x

^{2}- 24x + 18.

Comparing with the general form of the quadratic function, f(x) = ax

^{2}+ bx + c, we get

a = 4.

Since a = 4, we first take out 4 common from the first two terms

We get, f(x) = 4(x

^{2}- 6x )+ 18.

Now we complete the square of the first two terms in the quadratic equation. To complete the square, we add and subtract the square of the half of the second term.

So we get

f(x) = 4(x

^{2}- 6x + 9 - 9) + 12

=4(x

^{2}- 6x + 9) - 36 + 12

Now we can combine the first three terms and last two term.

We get,

**f(x) = 4(x - 3)**

^{2}- 24, which is the required standard form.The vertex of the above equation is (3, -24)