How to Convert to Vertex Form in Easy Steps

Does converting a quadratic equation to the vertex form leave you scratching your head? In this article, we explain how to convert to vertex form in easy steps, by hand, on your calculator, or using an online tool.

What is the vertex form?

The vertex form relates to the parabola of an equation. A parabola is a curvilinear graph that is produced by a quadratic function. Parabola distinctive characteristics that you can describe mathematically. These include: 

• The focus is a fixed point on the parabola.
• The directrix, a fixed line running perpendicular to the axis of the parabola. 
• The vertex is the point at which the curve intersects the axis of the graph – either its lowest or highest point. 

So, the vertex form of the equation characterizes the point of crossing of the axis of symmetry in a parabola. This form technically contains the information you’d need to draft the parabola of the quadratic equation, much like plotting coordinates. The vertex form of an equation has a specific format that contains the vertex details:

y = a·(x-h)² + k.

In a vertex quadratic equation, the letter a tells us whether the curve will open upwards or downwards. h and k specify the location of the vertex (x- and y- coordinates respectively) if you were to make a graph of this equation. 

Why is the vertex form important?

The parabola is a seminal mathematical concept that has a wide range of applications. Pascal considered it to be the projection of a circle, while Galileo related the parabola to the path taken as a projectile falls under uniform gravity. 

Parabolic paths are common in physics and both vertex and quadratic equations are used to calculate areas within an enclosed space, the speed, and the trajectory of physical bodies in motion, and to create precision curves for design. 

Here is a brief video that can help explain more about what vertex form is:

How to Convert to Vertex 

In summary, you can convert a standard equation into its vertex form by completing the square. Here is a step-by-step outline of how you complete the square and then convert the equation to vertex: 

Here is a standard equation to be converted:

y = -3x² – 6x – 9

To convert this equation to its vertex form, you need to deal with the “(a+b)² whole square” that is in the standard form. To convert the standard form as explained in the steps below, we’ll be completing the square.

Step 1: Identify the coefficient of x in the standard equation

If the coefficient of x² is not one, that factor needs to be placed outside as a common factor. In the case of the above equation, it is -3.

Step 2: Move the coefficient outside the formula

The coefficient is the value that moves outside the formula:

y = −3x² − 6x − 9 = −3 (x² + 2x + 3)

Now the coefficient of x² is 1 and you can readily convert this equation to its vertex form. 

Step 3: Complete the square

Identify the coefficient of x in the modified equation. In this equation, it is now 2. Half the coefficient and square the value. Here, the resulting value is 1. 

y = −3 (x2 + 2x + 3)

(2/2)²   = 1² = 1

Step 4: Add and subtract the value from step 3 after the x in the formula

Move your value from completing the square into the equation following the x value in your formula.

y = −3 (x² + 2x 1 – 1 + 3)

Step 5: Factorize the equation

Factorization is simply breaking up the equation into several simpler factors. Here, we’ll use the perfect square trinomial of the first three terms. 

y = −3 (x² + 2x 1 – 1 + 3)

becomes 

y = −3 ((x+1)² – 1 + 3)

Step 6: Simplify the formula again to obtain the vertex form

You’re almost there. All that needs to be done now is to further factorize the formula until you get a tidy vertex equation.

y = −3 ((x+1)² – 1 + 3)

Becomes

y = -3 ((x+1)² + 2)

Becomes

y = -3(x+1)² – 6  

This is the vertex form of the original equation. Below it is presented in the vertex format: 

y = a (x – h) 2 + k

For this example equation h = -1 and k = -6

You can use the vertex formula to identify vertex coordinates

Converting quadratic equations to the vertex form is a method for finding the coordinates of the vertex of the equation’s curve. If you are converting a standard equation to vertex form for this reason, you can derive the vertex coordinates (h and k) using an alternative method. Here is how to do it: 

Step 1: Take the standard form of a parabola and identify a, b, and c 

Here is the standard form of a parabola: 

y = ax² + bx + c

By using a, b, and see you don’t have to deal with a square root. 

Step 2: Use the formula below to calculate the h value

The h value is the x-coordinate of the parabola vertex. You can calculate this value with the following formula:

h = -b/(2a)

For our example formula: y = -3x² – 6x – 9 

a = -3, b = 6, and c = 9

So, 

h = 6 / (2 x -3)

Therefore

h = -1

Step 3: Use the formula below to calculate the k value

The h value is the x-coordinate of the parabola vertex. You can calculate this value with the following formula:

k = c – b²/(4a)

For our example formula: y = -3x² – 6x – 9

a = -3, b = 6, and c = 9

So,

k = -9 –  6² / 4(-3)

Therefore

k = -6

Use a graphing calculator to find the vertex

You can use a graphing calculator to find the h and k values you can then use to present the vertex form of an equation. Here are the basic steps for finding the vertex on a graphing calculator, using the example equation we worked through above: 

Step 1: Enter the equation on your graphing calculator 

You can input the equation in the expression panel. You may have to press the expression key which usually has ∱,  ∱₁, or even y= on it. Type in the formula using the keys on your calculator’s keypad. Once you have entered the formula, press Enter.

Step 2: Press the Graph button

Once you have entered your equation, press the graph button or symbol. The calculator will generate the graph of the equation. Most graphing calculators display the point at which the line of the graph crosses the y-axis, known as the y-intercept.

At this point, you may see the vertex graphically displayed and note its values. 

Step 3: Press calc to obtain a function analysis

Finding out the characteristics of this curve will proved us with the h and k values that determine the vertex. Press the calc or analysis button to bring up the characteristics of the curve.

Step 4: Look for the maximum or minimum to obtain your vertex values

The maxima or minima points are the point(s) at which the graph changes direction. The vertex will always be either a maximum or minimum of the equation you programmed your calculator with. 

As you can see, with this equation the h and k values are clearly maxima points. This works on any quadratic equation. 

Convert the standard equation to vertex using an online calculator

You can bypass the notepad and calculator and use an online calculator to derive the vertex form of your equation. There are many calculators online. Here is how we get the vertex form using the calculator at Omni calculator: 

Step one: Take the a, b, and c values from your standard formula

In our original equation y = -3x² – 6x – 9

So,

a = -3, b = -6, and c = -9

Step two: Enter the a, b, and c values in the fields of the online calculator

The Omni calculator will automatically plot the parabola and display the vertex graphically.

Step three: Scroll down to find the vertex conversion of your original formula

Beneath the graph, the calculator provides the key characteristics of the parabola, just like the graphing calculator. They provide the standard and vertex forms of the equation along with vertex and Y-intercept. This works for any quadratic equation. 

Final Thoughts

As you can see, obtaining the vertex form of a standard quadratic equation is not as mysterious as you thought. By working interchangeably between standard and vertex form equations, you can define the characteristics of virtually any parabola. Test these methods with your own equations, and remember, practice makes perfect!

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