The vertex form of a parabola's equation is generally expressed as: $ y = a(x-h)^2 +k $

(h,k) is the vertex as you can see in the picture below

If a is positive then the parabola opens upwards like a regular "U".

If a is negative, then the graph opens downwards like an upside down "U".

And, just like standard form, the larger the $ |a|$, the more narrow the parabola's graph gets.

The role of 'a'

Case I : When $|a| > 1 $

The larger the $ |a|$ is, the more the graphs narrows.

Case II : When $|a| < 1 $

The larger the $ |a|$ is, the more narrow the parabola is. Or, another way to think of it, the closer that the value of $a$ gets to zero, the wider the parabola becomes.

Vertex Form Practice Problems

Problem 1

What is the graph of the following parabola y = (x–1)² + 1?

The parabola's vertex is the point (1,1).

Problem 2

What is the graph of the following parabola y = –(x–1)² + 1?

Problem 3

What is the graph of the following parabola y = (x+2)² –3?

Identifying the vertex in vertex form

Problem 4.1

What is the vertex of the following parabola: y = (x + 3)² + 4