How to Write a Quadratic Function in Standard Form

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
• 73x² + 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,                            

ax² + bx + c = a (x – h)² + k

ax² + bx + c = a (x² – 2xh + h²) + k

ax² + bx + c = ax² – 2ah x + (ah² + k)

Comparing the coefficients of x on both sides,

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

Now comparing constants on both sides we get,

c = ah² + k

c = a (-b/2a)² + k (From (1))

c = b²/(4a) + k

k = c – (b²/4a)

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 – 916 =15116

Therefore, f(x) =  a(x – h)² + k = 0 

= a(x – (-b)2a)² + 4ac – b² 4a= 0

= 4(x – (-3)8)² + 15116 = 0

= 4(x + 38)² + 15116 = 0    

Practice Question: Q. Rewrite quadratic function in standard form: 2 (x² – 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 (x² – x – x + 1) + 11 = 0

3 (x² – 2x + 1) + 11 = 0

3x² – 4x + 2 + 11 = 0

3x² – 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 4x² + 6x – 18 = 0. 

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

4x² + 6x – 18 = 0

2(2x² + 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

4x² + 12x – 6x – 18 = 0

4x² + 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.