Vertex Form of a Quadratic Equation

In this video, we are going to look at the vertex form of a quadratic equation.
If we look at a regular quadratic function such as $y=x^2$, and want to shift it to the right 3 units and down 4 units, then we know that the modified equation would be
$y=(x-3)^2-4$

*Horizontal shifts are located within the parenthesis and vertical shifts are located outside of the parenthesis*

From this equation we can clearly see that the vertex point is at (3,-4).

This form is very useful in identifying the coordinates of the vertex.

If we were given an equation in standard form, we can complete the square to get it to vertex form.
For example:
$y=x^2+6x+8$
To solve by completing the square, we want to solve for the number to add by using
$(\frac{b}{2})^2$
So:
$(\frac{6}{2})^2=3^2=9$
This means that we have to add 9 to complete the square. However, if we add 9 then it changes the formula. To counteract that, we will also subtract 9.

Lets focus on the first three terms in the equation.
$y=x^2+6x+9+8-9$
When we factor those terms, we get
$y=(x+3)(x+3)+8-9$
or
$y=(x+3)^2-1$
Now, we have written the equation in vertex form. From here, we can see that the vertex is at (-3,-1).

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Used by students across the country. Pre-Algebra, Algebra I, Geometry, & Algebra II