Standard Form to Vertex Form

In this mini-lesson, we will explore the process of converting standard form to vertex form and vice-versa. The standard form of a parabola is y = ax 2 + bx + c and the vertex form of a parabola is y = a (x - h) 2 + k. Here, the vertex form has a square in it. So to convert the standard to vertex form we need to complete the square.

Let us learn more about converting standard form to vertex form along with more examples.

Standard Form and Vertex Form of a Parabola

The equation of a parabola can be represented in multiple ways like: standard form, vertex form, and intercept form. One of these forms can always be converted into the other two forms depending on the requirement. In this article, we are going to learn how to convert

• standard form to vertex form and
• vertex form to standard form

Let us first explore what each of these forms means.

Standard Form

The standard form of a parabola is:

• y = ax 2 + bx + c

Here, a, b, and c are real numbers (constants) where a ≠ 0. x and y are variables where (x, y) represents a point on the parabola.

Vertex Form

The vertex form of a parabola is:

• y = a (x - h) 2 + k

Here, a, h, and k are real numbers, where a ≠ 0. x and y are variables where (x, y) represents a point on the parabola.

How to Convert Standard Form to Vertex Form?

In the vertex form, y = a (x - h) 2 + k, there is a "whole square". So to convert the standard form to vertex form, we just need to complete the square. But apart from this, we have a formula method also for doing this. Let us look into both methods.

By Completing the Square

Let us take an example of a parabola in standard form: y = -3x 2 - 6x - 9 and convert it into the vertex form by completing the square. First, we should make sure that the coefficient of x 2 is 1. If the coefficient of x 2 is NOT 1, we will place the number outside as a common factor. We will get:

y = −3x 2 − 6x − 9 = −3 (x 2 + 2x + 3)

Now, the coefficient of x 2 is 1. Here are the steps to convert the above expression into the vertex form.

Step 1: Identify the coefficient of x.

Step 2: Make it half and square the resultant number.

Step 3: Add and subtract the above number after the x term in the expression.

Step 4: Factorize the perfect square trinomial formed by the first 3 terms using the suitable identity.

Here, we can use x 2 + 2xy + y 2 = (x + y) 2 .

In this case, x 2 + 2x + 1= (x + 1) 2

The above expression from Step 3 becomes:

Step 5: Simplify the last two numbers and distribute the outside number.

Here, -1 + 3 = 2. Thus, the above expression becomes:

This is of the form y = a (x - h) 2 + k, which is in the vertex form. Here, the vertex is, (h, k)=(-1,-6).

By Using the Formula

In the above method, ultimately we could find the values of h and k which are helpful in converting standard form to vertex form. But the values of h and k can be easily found by using the following steps:

• Find h using h = -b/2a.
• Since (h, k) lies on the given parabola, k = ah 2 + bh + c. Just use this to find k by substituting the value of 'h' from the above step.

Let us convert the same example y = -3x 2 - 6x - 9 into standard form using this formula method. Comparing this equation with y = ax 2 + bx + c, we get a = -3, b = -6, and c = -9. Then

(i) h = -b/2a = -(-6) / (2 × -3) = -1

(ii) k = -3(-1) 2 - 6(-1) - 9 = -3 + 6 - 9 = -6

Substitute these two values (along with a = -3) in the vertex form y = a (x - h) 2 + k, we get y = -3 (x + 1) 2 - 6. Note that we have got the same answer as in the other method.

Which method is easier? Decide and go ahead.

Tips and Tricks:

If the above processes seem difficult, then use the following steps:

• Compare the given equation with the standard form (y = ax 2 + bx + c) and get the values of a,b, and c.
• Apply the following formulas to find the values the values of h and k and substitute it in the vertex form (y = a(x - h) 2 + k):
  h = -b/2a
  k = -D/4a

Here, D is the discriminant where D = b 2 - 4ac.

How to Convert Vertex Form to Standard Form?

To convert vertex form into standard form, we just need to simplify a (x - h) 2 + k algebraically to get into the form ax 2 + bx + c. Technically, we need to follow the steps below to convert the vertex form into the standard form.

• Expand the square, (x − h) 2 .
• Distribute 'a'.
• Combine the like terms.

Example: Let us convert the equation y = -3 (x + 1) 2 - 6 from vertex to standard form using the above steps:

y = -3 (x + 1) 2 - 6
y = -3 (x + 1)(x + 1) - 6
y = -3 (x 2 + 2x + 1) - 6
y = -3x 2 - 6x - 3 - 6
y = -3x 2 - 6x - 9

Important Notes on Standard Form to Vertex Form:

• In the vertex form, (h, k) represents the vertex of the parabola where the parabola has either maximum/minimum value.
• If a > 0, the parabola has the minimum value at (h, k) and
  if a < 0, the parabola has the maximum value at (h, k).

☛ Related Topics:

• Vertex Calculator
• Quadratic Function Calculator

Standard Form to Vertex Form Examples

Example 1: Find the vertex of the parabola y = 2x 2 + 7x + 6 by completing the square. Solution: The given equation of parabola is y = 2x 2 + 7x + 6. To find its vertex, we will convert it into vertex form. To complete the square, first, we will make the coefficient of x 2 as 1. We will take the coefficient of x 2 (which is 2 in this case) as the common factor. 2x 2 + 7x + 6 = 2 (x 2 + 7/2 x + 3) The coefficient of x is 7/2, half it is 7/4, and its square is 49/16. Adding and subtracting it from the quadratic polynomial that is inside the brackets of the above step, 2x 2 + 7x + 6 = 2 (x 2 + 7/2 x + 49/16 - 49/16 + 3) Factorizing the quadratic polynomial x 2 + 7/2 x + 49/16, we get (x + 7/4) 2 . Then 2x 2 + 7x + 6 = 2 ((x + 7/4) 2 - 49/16 + 3) = 2 ((x + 7/4) 2 - 1/16) = 2 (x + 7/4) 2 - 1/8 By comparing this with a (x - h) 2 + k, we will get (h, k) = (-7/4, -1/8). Answer: The vertex of the given parabola is (-7/4, -1/8).

Example 2: Find the vertex of the same parabola as in Example 1 without converting into vertex form. Solution: The given parabola is y = 2x 2 + 7x + 6. So a = 2, b = 7, and c = 6. The x-coordinate of the vertex is, h = -b/2a = -7/[2(2)] = -7/4. The y-coordinate of the vertex is, k = 2(-7/4) 2 + 7(-7/4) + 6 = -1/8 Answer: We have got the same answer as in Example 1 which is (h, k) = (-7/4, -1/8).

Example 3: Find the equation of the following parabola in standard form. Solution: We can see that the parabola has the maximum value at the point (2, 2). So the vertex of the parabola is, (h, k) = (2, 2). So the vertex form of the above parabola is, y = a (x - 2) 2 + 2 . (1). To find 'a' here, we have to substitute any known point of the parabola in this equation. The graph clearly passes through the point (1, 0) = (x, y). Substitute it in (1): 0 = a (1 - 2) 2 + 2
0 = a + 2
a = -2 Substitute it back into (1) and expand the square to convert it into the standard form: y = -2 (x - 2) 2 + 2
y = -2 (x 2 - 4x + 4) + 2
y = -2x 2 + 8x - 8 + 2
y = -2x 2 + 8x - 6 Answer: Thus, the standard form of the given parabola is: y = -2x 2 + 8x - 6.

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