Put Parabola in Standard Form Calculator

What is Standard Form of a Parabola?

The standard form of a parabola is written as:

Where:

• (h, k) is the vertex of the parabola
• p is the distance from the vertex to the focus
• The parabola opens upwards or to the right if p is positive, and downwards or to the left if p is negative

The standard form makes it easy to identify key features of the parabola, such as its vertex, focus, and directrix.

How to Convert a Parabola to Standard Form

Converting a parabola to standard form involves completing the square for the x or y terms, depending on whether the parabola opens vertically or horizontally.

For Vertical Parabolas (opens up or down)

1. Start with the general form: y = ax² + bx + c
2. Factor out the coefficient of x² from the first two terms: y = a(x² + (b/a)x) + c
3. Complete the square inside the parentheses:
   • Take half of the coefficient of x, square it, and add and subtract it inside the parentheses
   • This gives: y = a[(x + (b/2a))² - (b²/4a²)] + c
4. Distribute the a and combine like terms to get the standard form

For Horizontal Parabolas (opens left or right)

1. Start with the general form: x = ay² + by + c
2. Factor out the coefficient of y² from the first two terms: x = a(y² + (b/a)y) + c
3. Complete the square inside the parentheses:
   • Take half of the coefficient of y, square it, and add and subtract it inside the parentheses
   • This gives: x = a[(y + (b/2a))² - (b²/4a²)] + c
4. Distribute the a and combine like terms to get the standard form

Examples of Converting to Standard Form

Let's look at a couple of examples to see how the conversion process works.

Example 1: Vertical Parabola

Convert y = 2x² + 8x + 3 to standard form.

1. Factor out the coefficient of x²: y = 2(x² + 4x) + 3
2. Complete the square:
   • Half of 4 is 2, squared is 4
   • Add and subtract 4 inside the parentheses: y = 2(x² + 4x + 4 - 4) + 3
   • This becomes: y = 2[(x + 2)² - 4] + 3
3. Distribute the 2: y = 2(x + 2)² - 8 + 3
4. Combine like terms: y = 2(x + 2)² - 5
5. Now, rewrite in standard form: (x + 2)² = (1/2)(y + 5)

Example 2: Horizontal Parabola

Convert x = 3y² - 6y + 2 to standard form.

1. Factor out the coefficient of y²: x = 3(y² - 2y) + 2
2. Complete the square:
   • Half of -2 is -1, squared is 1
   • Add and subtract 1 inside the parentheses: x = 3(y² - 2y + 1 - 1) + 2
   • This becomes: x = 3[(y - 1)² - 1] + 2
3. Distribute the 3: x = 3(y - 1)² - 3 + 2
4. Combine like terms: x = 3(y - 1)² - 1
5. Now, rewrite in standard form: (y - 1)² = (1/3)(x + 1)