vertex form and identify the vertex

Question

f(x)=x²-4x+9

Answer 1

You can get the vertex form by completing the square.

this is currently in standard form (ax² + bx + c)

In f(x) = x²-4x+9, b = -4

We need to get (b/2)² to complete the square. 

b = -4

(b/2)² = (-4/2)² = (-2)² = 4

Now we take this number and add it to both sides of the equation.

f(x) + 4 = x² - 4x + 9 + 4

f(x) + 4 = x² - 4x + 4 + 9                 We rearrange the equation.

f(x) + 4 = (x² - 4x + 4) + 9             Now we can separate the perfect square.

f(x) + 4 = (x-2)² + 9                         We can factor the highlighted area.

f(x) = (x-2)² + 9 - 4                          Now we solve for f(x)

f(x) = (x-2)² + 5                           This is the vertex form of that equation.

 

The vertex form is in the format of y = a(x-h)² + k where the vertex is (h, k).

In this case, h = 2 and k = 5 so the vertex is (2,5)