Standard and Vertex Form of a Parabola

The standard form and vertex form are two different ways to express the equation of a parabola. Converting between these forms is useful for solving mathematical problems and performing statistical analysis, depending on which form makes the task easier.

Standard Form

y = ax² + bx + c

x = ay² + by + c

Here, 

• (0, 0) is the vertex
• a, b, and c are constants, and a ≠ 0

If a > 0, the parabola opens upwards, and if a < 0, the parabola opens downwards.

The standard forms are used to identify the direction in which the parabola opens and determine the y-intercept (c). 

Vertex Form

y = a(x – h)² + k

x = a(y – k)² + h

Here,

• (h, k) is the vertex
• a is the constant that determines the direction of the parabola.

The parabola opens upwards if a > 0, and it opens downwards if a < 0 

The vertex forms are used to identify the vertex (h, k) of the parabola, making it easier to graph and determine its maximum or minimum point. 

Standard Form to Vertex Form

Equations in standard form can be converted to their corresponding vertex form either by completing the square method or by using the vertex.

By Completing the Square

Let us convert the parabola y = 2x² + 8x + 7 to its vertex form by the method of completing the square.

Factoring out the GCF 

⇒ y = 2(x² + 4x) + 7

Simplifying

To complete the square, we add and subtract 8 on the right side of the equation.

⇒ y = 2(x² + 4x) + 7 + 8 – 8

⇒ y = 2(x² + 4x + 4) + 7 – 8

⇒ y = 2(x + 2)² – 1

Thus, the equation in vertex form is y = 2(x + 2)² – 1….(i), which is of the form y = a(x – h)² + k ….(ii)

Now, on comparing equation (i) with (ii), we get

a = 2, h = -2, and k = -1 and the vertex is at (-2, -1)

By Using the Vertex 

As we know, in the standard form of the parabola y = ax² + bx + c, the vertex (h, k) is given by the formula:

• h = ${-\dfrac{b}{2a}}$
• k = f(h) = ah² + bh + c, since (h, k) lies on the parabola

Now, let us convert y = 2x² + 8x + 7 to its vertex form by using these formulas.

Comparing 

Comparing the equation with the standard form of the parabola y = ax² + bx + c, we get

a = 2, b = 8, and c = 7 …..(i)

Finding the Vertex

h = ${-\dfrac{b}{2a}}$ = ${-\dfrac{8}{2\times 2}}$ = -2

k = f(-2) = 2(-2)² + 8(-2) + 7 = -1

Thus, the vertex is at (-2, -1) …..(ii)

Substituting 

Now, substituting the values of (i) and (ii) in the vertex form y = a(x – h)² + k, we get

y = 2(x + 2)² – 1

Thus, the equation in vertex form is y = 2(x + 2)² – 1

Vertex Form to Standard Form

Now, we will convert the parabola y = 2(x + 2)² – 1 to its corresponding standard form.

Expanding

y = 2(x + 2)(x + 2) – 1

⇒ y = 2(x² + 4x + 4) – 1

Simplifying

⇒ y = 2x² + 8x + 8 – 1

⇒ y = 2x² + 8x + 7….(i), which is of the form y = ax² + bx + c ….(ii)

Now, on comparing equation (i) with (ii), we get

a = 2, b = 8, and c = 7, and the vertex is at (0, 0)

Solved Examples