Put Quadratic Function in Standard Form Calculator

A quadratic function is a second-degree polynomial that can be written in several forms. The standard form is particularly useful for analyzing the function's properties, such as its vertex, axis of symmetry, and direction of opening.

What is Standard Form?

The standard form of a quadratic function is written as:

f(x) = a(x - h)² + k

where (h, k) is the vertex of the parabola, and a determines the direction and width of the parabola.

This form is called the vertex form because it clearly shows the vertex of the parabola. The standard form is equivalent to the vertex form, but it's written as:

f(x) = ax² + bx + c

The standard form is useful for finding the roots of the quadratic equation, which are the x-intercepts of the parabola.

How to Convert to Standard Form

To convert a quadratic function from its given form to standard form, follow these steps:

Identify the coefficients a, b, and c in the given quadratic function.
If the function is in vertex form, convert it to standard form by expanding the squared term.
If the function is in factored form, multiply the factors to get the standard form.
Simplify the expression to get the final standard form.

Note: The standard form is not unique. Different forms can represent the same quadratic function. However, the standard form is the most commonly used form for analyzing quadratic functions.

Worked Example

Let's convert the quadratic function f(x) = 2x² - 8x + 3 to standard form.

Step 1: Identify the coefficients. Here, a = 2, b = -8, and c = 3.

Step 2: The function is already in standard form, so no further conversion is needed.

The standard form of the quadratic function is:

f(x) = 2x² - 8x + 3

FAQ

What is the standard form of a quadratic function?: The standard form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants.
How do I convert a quadratic function to standard form?: To convert a quadratic function to standard form, identify the coefficients a, b, and c in the given function. If the function is in vertex form, convert it to standard form by expanding the squared term. If the function is in factored form, multiply the factors to get the standard form.
What is the vertex form of a quadratic function?: The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola, and a determines the direction and width of the parabola.
What is the difference between standard form and vertex form?: The standard form is written as f(x) = ax² + bx + c, while the vertex form is written as f(x) = a(x - h)² + k. The standard form is useful for finding the roots of the quadratic equation, while the vertex form is useful for analyzing the function's properties, such as its vertex, axis of symmetry, and direction of opening.
What is the standard form of a quadratic equation?: The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants.