Put Quadratic Function in Standar Form Calculator

A quadratic function in standard form is written as f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify key features of the quadratic function.

What is Standard Form?

The standard form of a quadratic function is f(x) = a(x - h)² + k. This form is called the vertex form because it clearly shows the vertex of the parabola at the point (h, k).

The standard form is particularly useful because:

It immediately reveals the vertex of the parabola
It shows whether the parabola opens upwards or downwards (depending on the sign of 'a')
It makes it easy to identify the axis of symmetry (x = h)

f(x) = a(x - h)² + k

Where:

a determines the parabola's width and direction (upwards if a > 0, downwards if a < 0)
(h, k) are the coordinates of the vertex

How to Convert to Standard Form

To convert a quadratic function from its general form (f(x) = ax² + bx + c) to standard form, follow these steps:

Factor out the coefficient of x² from the first two terms
Complete the square for the expression inside the parentheses
Distribute the coefficient of x² back through the completed square
Add and subtract the constant term to move it outside the squared term

f(x) = ax² + bx + c
= a(x² + (b/a)x) + c
= a[(x² + (b/a)x + (b/2a)²) - (b/2a)²] + c
= a(x + b/2a)² - (b²/4a) + c
= a(x - h)² + k

Where h = -b/2a and k = c - b²/4a

Note: The constant term in standard form is k = c - b²/4a, not just c. This is because completing the square changes the constant term.

Worked Example

Let's convert f(x) = 2x² + 8x + 5 to standard form:

Factor out the coefficient of x² from the first two terms:
f(x) = 2(x² + 4x) + 5
Complete the square inside the parentheses:
x² + 4x = (x² + 4x + 4) - 4 = (x + 2)² - 4
Substitute back and distribute the 2:
f(x) = 2[(x + 2)² - 4] + 5 = 2(x + 2)² - 8 + 5
Combine the constants:
f(x) = 2(x + 2)² - 3

The standard form is f(x) = 2(x + 2)² - 3, with vertex at (-2, -3).

FAQ

Why is standard form useful?

The standard form of a quadratic function makes it easy to identify key features like the vertex, the direction the parabola opens, and the axis of symmetry. This information is not immediately obvious from the general form.

How do I know if a quadratic function is in standard form?

A quadratic function is in standard form if it is written as f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola.

What happens if the coefficient of x² is negative?

If the coefficient 'a' is negative, the parabola will open downwards instead of upwards. The standard form still applies, but the parabola's direction is determined by the sign of 'a'.

Can I convert any quadratic function to standard form?

Yes, any quadratic function in the form f(x) = ax² + bx + c can be converted to standard form using the completing the square method.