Quadratic Functions are defined as second-degree polynomial equation, which means it has at least one term with a power of two. Quadratic Functions are so named because Quad stands for ‘four’ (squared), and a quadratic function’s greatest degree should be 2.

**Quadratic Functions can be represented in 3 forms:**

- Standard Form :
**ax² + bx + c = 0** - Vertex Form :
**a(x – h)² + k = 0** - Intercept Form :
**a(x – p)(x – q) = 0**

**How to write a quadratic function in standard form?**

**This is how to write the quadratic function in standard form:**

**f(x) = ax² + bx + c = 0**

Here a, b, and c are the constant coefficients and x is the unknown variable with the highest degree of 2, a is never equal to zero, making f(x) a quadratic function. The leading coefficient is always a non-zero real number, and it is denoted by ‘a.’ Otherwise, the function will not be quadratic since the greatest degree of 2 will not exist. The standard form of a quadratic function is also referred to as the general form of a quadratic function.

**Examples of Standard Form of Quadratic Equation:**

**4x² + 3x + 10 = 0****-x² + 5x = 0****7****3****x² + 7x + 3 = 0****9x² – 1 = 0**

**Changing a Quadratic Equation from Standard Form to Vertex Form**

Vertex form of a quadratic function is:

**f(x) = a(x – h)² + k = 0**

where, (h,k) is the vertex of the quadratic function f(x).

and, a is the coefficient of x².

Therefore, equating a Standard Quadratic function and Vertex Quadratic function

we get,

ax^{2} + bx + c = a (x – h)^{2} + k

ax^{2} + bx + c = a (x^{2} – 2xh + h^{2}) + k

ax^{2} + bx + c = ax^{2} – 2ah x + (ah^{2} + k)

Comparing the coefficients of x on both sides,

b = -2ah or **h = ****(-b)/****2a**

Now comparing constants on both sides we get,

c = ah^{2} + k

c = a (-b/2a)^{2} + k (From (1))

c = b^{2}/(4a) + k

k = c – (b^{2}/4a)

**k = ****4ac – b² ****4a**

Now we can rewrite Vertex Form into Standard Form as:

**f(x) = a(x – ****(-b)****2a****)² + ****4ac – b² ****4a****= 0**

For Example, For a Standard Quadratic Function f(x) = 4x² + 3x + 10 = 0

Vertex Form will be,

a = 4

h = **(-b)****2a**** = ****(-3)****2×4**** = **-38

k =** ****4ac – b² ****4a**** **= **4x4x10 – 3² ****4×4**** **= **160 – 9****16**** =****151****16**

Therefore, f(x) = a(x – h)² + k = 0

= a(x – (-b)2a)² + 4ac – b² 4a= 0

= 4(x – (-3)8)² + **151****16**** **= 0

= **4(x + ****3****8****)² + ****151****16**** = 0**

Practice Question: Q. Rewrite quadratic function in standard form: 2 (x^{2} – 2x + 1) + 1 = 0

**Changing from Vertex Form to Standard Form**

By evaluating and simplifying (x – h)^{2} = (x – h) (x – h), a quadratic equation can be converted from its vertex form to its standard form:

For Example,

3 (x – 1)^{2} + 11 = 0 is in Vertex Form

Therefore by substituting (x – h)^{2} = (x – h) (x – h),

3 (x – 1) (x – 1) + 11 = 0

3 (x^{2} – x – x + 1) + 11 = 0

3 (x^{2} – 2x + 1) + 11 = 0

3x^{2 }– 4x + 2 + 11 = 0

**3x**^{2}** – 4x + 13 = 0**, which is in Standard Form.

**Changing a Quadratic Equation from Standard to Intercept Form**

A (x – p)(x – q) = 0 is used to transform the standard form of a quadratic equation to the vertex form.

The x-intercepts of the quadratic function f(x) = ax² + bx + c = 0 are (p, 0) and (q, 0), respectively, therefore p and q are the roots of the quadratic equation.

To find p and q, we simply utilize any of the quadratic equation solving methods.

For example,

For the quadratic equation 4x^{2} + 6x – 18 = 0.

We’ll now factorize the quadratic equation to solve it.

4x^{2} + 6x – 18 = 0

2(2x^{2} + 3x – 9) = 0

2(2x – 3) (x + 3) = 0

2x – 3 = 0 and x + 3 =0

2x = 3; x + 3 = 0

Hence, **a= 4**, **p = 3/2** and **q = -3**

Thus, the intercept form is,

a (x – p)(x – q) = 0

2 (x – 3/2) (x – (-3)) = 0

2 (x – 3/2) (x + 3) = 0

**(2x – 3) (2x + 6) = 0**

**Changing from Intercept Form to Standard Form**

A quadratic equation can be transformed from its intercept form to its standard form by multiplying and simplifying (x – p) (x – q):

For Example,

(2x – 3) (2x + 6) = 0 is in Intercept Form

4x^{2} + 12x – 6x – 18 = 0

**4x**^{2}** + 6x – 18 = 0**, which is in Standard Form.

**Graphing Quadratic Functions in Standard Form**

A quadratic function is defined as f(x) = ax2 + bx + c, where a, b, and c are all non-zero values. A parabola is a curve that represents the graph of a quadratic function.

Different types of parabolas can have different widths and slopes but the basic U structure always remains the same. The axis of symmetry of a parabola crosses at the vertex of the parabola. All parabolas are symmetric because of this line known as the axis of symmetry.

The quadratic function **f(x) = ax² + bx + c = 0** is said to be in standard form, wherein a (coefficient of x²) is not equal to zero. If a is positive, the parabola of the graph will open upward but if it is negative, it will open downward.

**Application of the Quadratic Function**

Quadratics and parabolas are used in a variety of real-life scenarios. Cases deal with the highest or lowest point (vertex of a parabola), and all deal with the concept of the quadratic function. Quadratic equation applications include Projectile motion, Velocity, acceleration, Geometry, etc.

**The following are some examples of such situations:**

- Architecture: Many projects reveal the use of parabolic figures to form the foundation of buildings, bridges, amusement parks, etc.
- Shooting a Canon: The canons are used to shoot heavy explosives, and determining the projectile angle(parabola) can use quadratics.
- Motion: Problems dealing with velocity, displacement, acceleration, etc., utilize Quadratic formulas widely.
- Geometry: Problems involving determining the area of different shapes such as rectangles, parallelograms, and so on are basic quadratic equation applications.
- Symmetry: The symmetry of parabolic figures helps develop a knowledge of the symmetry of other forms. The use of quadratics to derive an equation can also assist with the comprehension of symmetry.

** **You can find the vertex if you know the equation for the function that models the situation. If

the function is f(x) = ax² + bx + c = 0

We know that the x-coordinate of the vertex will be **(-b)****2a**** **and the y-coordinate of the vertex can be found by substituting the x-coordinate into the function.

A quadratic function is a polynomial equation with a maximum degree of two. The variable’s power is always a positive integer less than or equal to two. Standard Form, Vertex Form, and Intercept form are the three ways to express a quadratic function.

When graphed, the standard form of the equation always produces a parabola. The value of the highest degree variable’s coefficient affects the shape of this parabola. Quadratic equations can answer a wide range of real-world problems. Some quadratic equation applications can be based on speed problems and Geometry area questions. Dealing with questions related to quadrilaterals such as distance, speed, time, etc.

**Frequently Asked Questions**

**Q1. What are Quadratic Functions?**

A second-degree polynomial equation contains at least one term with a power of two, which is what quadratic functions are. Quadratic Functions are so named because Quad stands for ‘Four’ (squared) and the largest degree of a quadratic function should be 2.

**Q2. How to write a quadratic function in a standard form?**

This is how to write a quadratic function in a standard form:

**f(x) = ax² + bx + c = 0**

f(x) is a quadratic function where a, b, and c are constant coefficients and x is the unknown variable with the greatest degree of 2, and a cannot be equal to zero. The leading coefficient, represented by ‘a,’ is always a non-zero real number. Otherwise, if the maximum degree of 2 does not exist, the function will not be quadratic.

**Q3. How many forms are there to represent a Quadratic Function?**

Quadratic functions can be expressed in three different ways:

- Standard Form :
**ax² + bx + c = 0** - Vertex Form :
**a(x – h)² + k = 0** - Intercept Form :
**a(x – p)(x – q) = 0**

**Q4. What type of graph is formed when a Quadratic Function is plotted?**

A parabola is a curve that depicts a quadratic function’s graph.

The widths and slopes of different types of parabolas can vary, but the basic U structure is always the same. A parabola’s axis of symmetry crosses at the parabola’s vertex. Due to this line, known as the axis of symmetry, all parabolas are symmetric.

**Q5. What effect does the coefficient of x² have on the parabola?**

Varied forms of parabolas have different widths and slopes, but the underlying U structure remains the same. When a (coefficient of x2) is not equal to zero, the quadratic function f(x) = ax2 + bx + c = 0 is said to be in standard form. If a is positive, the graph’s parabola will open upward; if it is negative, the graph’s parabola will open downward.

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