# Vertex form

Vertex form is another form of a quadratic equation. The standard form of a quadratic equation is ax^{2} + bx + c. The vertex form of a quadratic equation is

a(x - h)^{2} + k

where a is a constant that tells us whether the parabola opens upwards or downwards, and (h, k) is the location of the vertex of the parabola. This is something that we cannot immediately read from the standard form of a quadratic equation. Vertex form can be useful for solving quadratic equations, graphing quadratic functions, and more.

The following are two examples of quadratic equations written in vertex form:

- 2(x - 7)
^{2}+ 3; vertex at (7, 3) - 2(x + 7)
^{2}- 3; vertex at (-7 , -3)

The above examples show that we can't just read off the values based on their position in the equation. We need to remember the vertex form a(x - h)^{2} + k. If, like in equation (1.) above, the signs in the equation match that of the generalized vertex form, then we can read off (h, k) as the vertex. However, if, like in equation (2.), the signs are different from those in the general vertex form equation, we need to take the signs into account; for h, the sign of the x-coordinate of the vertex is opposite of that in the vertex form equation; for k, the sign of the y-coordinate is the same as that in the vertex form equation. This is due to the nature of positive/negative numbers.

## Converting from standard form to vertex form

Converting a quadratic equation from standard form to vertex form involves a technique called completing the square. Refer to the completing the square for a detailed explanation. Generally, it involves moving the constant to the other side of the equation and finding a constant that allows us to write the right hand side of the equation in a form resembling vertex form, applying that constant to the left side of the equation, then shifting the constant on the left side back to the right side.

Example

Convert y = 3x^{2} + 9x + 4 to vertex form:

y - 4 = 3x^{2} + 9x

y - 4 = 3(x^{2} + 3x)

y - 4 + 3(?) = 3(x^{2} + 3x + ?)

y - 4 + 3() = 3(x^{2} + 3x + )

y - = 3(x + )^{2}

y = 3(x + )^{2} -

This is our equation in vertex form, which tells us that the vertex is at (, ), and also that our parabola opens upwards, since a (3 in this case) is positive.

To convert from vertex form to standard form, we simply expand vertex form. We can confirm that our above equation in vertex form is the same as the original equation in standard form by expanding it:

y = 3(x + )^{2} -

y = 3(x^{2} + x + x + ()^{2}) -

y = 3(x^{2} + 3x + ) -

y = 3x^{2} + 9x + -

y = 3x^{2} + 9x +

y = 3x^{2} + 9x + 4

Expanding our equation in vertex form yields the same equation we started with in standard form, so we've confirmed that our conversion to vertex form was correct.