Quadratic Equations: Standard and Vertex Form
Unlike linear equations, which graph as lines, quadratic equations graph as parabolas (which look like a lot like Us or upside down Us). There is some important vocabulary to learn about parabolas:
There are several ways to write a quadratic equation and each form gives you an "easy" way to identify some facts about a parabola.

$x^2+6x+9$

 To convert from standard to vertex form, you can plug in $(h,k)$, which you can derive from $\dfrac{b}{2a}$, and $a$ into vertex form.
$\eqalign{y&=x^24x+6\\\text{vertex}=(x,y), x&=\dfrac{b}{2a}=\dfrac{4}{2}=2\\\text{Plug x value into equation to find y of vertex}\\y&=2^24(2)+6\\y&=48+6\\y&=2\\\text{vertex}&=(2,2), h=2, k=2\\\text{Plug vertex in as (h,k)}\\y&=a(xh)^2+k\\y&=1(x2)^2+2\\y&=(x2)^2+2}$
 To convert from vertex to standard form, use $(h,k)$ and $a$ to solve for $b$ (using $\dfrac{b}{2a}$) and then $c$.
$\eqalign{y&=2(x+2)^23\\\text{Find vertex and plug x and a into vertex formula to solve for b}\\(h,k)&=(2,3), a=2\\2&=\dfrac{b}{2a}\\2&=\dfrac{b}{2(2)}\\2&=\dfrac{b}{4}\\8&=b\\b&=8\\\text{Plug a and b into standard form}\\y&=2x^2+8x+c\\\text{Plug any point (like vertex) in for x and to solve for c}\\3&=2(2)^2+8(2)+c\\3&=8+c\\c&=5\\y&=2x^2+8x+5}$
Practice Problems:
Quadratic Equations: Standard and Vertex Form
Find the vertex of each equation:
1. $y=3x^26x+5$
2. $y=x^2+8x3$
3. $y=2x^2x+1$
4. $y=5(x7)^23$
5. $y=2.5(x+2)^2+1$
6. $y=2(x6)^24$
Rewrite each of the following equations in vertex and standard form.
7. vertex (1,2), point (2, 5)
8. vertex (3,6), yintercept 2