How to find the vertex form of a quadratic function

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 = ax2 + 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 = ax2 + 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 = -3x2 - 6x - 9 and convert it into the vertex form by completing the square. First, we should make sure that the coefficient of x2 is 1. If the coefficient of x2 is NOT 1, we will place the number outside as a common factor. We will get:

y = −3x2 − 6x − 9 = −3 (x2 + 2x + 3)

Now, the coefficient of x2 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 x2 + 2xy + y2 = (x + y)2.

In this case, x2 + 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 = ah2 + 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 = -3x2 - 6x - 9 into standard form using this formula method. Comparing this equation with y = ax2 + 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 = ax2 + 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 = b2 - 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 ax2 + 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 (x2 + 2x + 1) - 6
y = -3x2 - 6x - 3 - 6
y = -3x2 - 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

1. Example 1: Find the vertex of the parabola y = 2x2 + 7x + 6 by completing the square.

   Solution:

   The given equation of parabola is y = 2x2 + 7x + 6. To find its vertex, we will convert it into vertex form.

   To complete the square, first, we will make the coefficient of x2 as 1.

   We will take the coefficient of x2 (which is 2 in this case) as the common factor.

   2x2 + 7x + 6 = 2 (x2 + 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,

   2x2 + 7x + 6 = 2 (x2 + 7/2 x + 49/16 - 49/16 + 3)

   Factorizing the quadratic polynomial x2 + 7/2 x + 49/16, we get (x + 7/4)2. Then

   2x2 + 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).

2. Example 2: Find the vertex of the same parabola as in Example 1 without converting into vertex form.

   Solution:

   The given parabola is y = 2x2 + 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).

3. 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 (x2 - 4x + 4) + 2
   y = -2x2 + 8x - 8 + 2
   y = -2x2 + 8x - 6

   Answer: Thus, the standard form of the given parabola is: y = -2x2 + 8x - 6.

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FAQs on Standard Form to Vertex Form

How Do You Convert Standard Form to Vertex Form?

To convert standard form to vertex form,

• Convert y = ax2 + bx + c into the form y = a (x - h)2 + k by completing the square.
• Then y = a (x - h)2 + k is the vertex form.

How Do You Convert Vertex Form to Standard Form?

Converting vertex form into standard form is so easy. Just expand the square in y = a (x - h)2 + k, then expand the brackets, and finally simplify.

How to Convert Standard Form to Vertex Form Using Completing the Square?

To convert standard form to vertex form by using completing the square method,

• Take the coefficient of x2 as the common factor if it is other than 1.
• Make the coefficient of x half and square it.
• Add and subtract this number to the quadratic expression of the first step.
• Then apply algebraic identities to write it in the vertex form.

How to Find the Vertex of a Parabola in Standard Form?

Vertex can't be directly identified from standard form. Convert standard form into vertex form y = a (x - h)2 + k, then (h, k) would give the vertex of the parabola.

How to Convert Standard Form to Vertex Form Without Completing the Square?

To convert y = ax2 + bx + c into y = a (x - h)2 + k without completing the square, just find 'h' and 'k' using the following formulas

• h = -b / 2a
• k = - (b2 - 4ac) / 4a

What is the Use of Converting Standard Form into Vertex Form?

Vertex form is more helpful in graphing quadratic functions where we can easily identify the vertex, and by finding one/two points on either side of the vertex would give the perfect shape of a parabola.

How do you find the vertex equation?

How do you do vertex form step by step?