Quadratic Functions   f(x) = a(x - h)2 + k

-- Section 3.2 --

  Write a Quadratic Function in Standard Form
  f(x) = x2 + 6x + 8

Solve

Step

Graph

f(x) = x2 + 6x + 8

Complete the square *

f(x) = x2 + 6x + 9 - 9 + 8 Take half of 6 and square it
f(x) = (x + 3)2 - 1 (x + 3)2 = x2 + 6x + 9

Vertex (-3, -1)

Vertex is (h,k)
This means the vertex is shifted 3 units left and 1 unit down from the origin.
*Check out  completing the square for help with this step.

Click here ....

     Write a Quadratic Function in Standard Form
  f(x) = x2 + 4x + 6

[Solution]

 Write a Quadratic Function in Standard Form
  f(x) = -x2 + 10x - 5

Solve

Step

Graph

f(x) = -(x2 - 10x) - 5

Factor out the negative sign first 

f(x) = -(x2 - 10x + 25 - 25) - 5 Now complete the square* inside parenthesis
Remove -25 from parenthesis 
f(x) = -(x2 - 10x + 25) + 25 - 5 Notice sign change

f(x) = -(x - 5)2  + 20

(x - 5)2 = x2 - 10x + 25

Vertex (5, 20)

Vertex is (h,k)
This means the vertex is shifted 5 units right and 20 units up from the origin.
*Check out  completing the square for help with this step.

Click here ....

 Write a Quadratic Function in Standard Form
                                f(x) = -x2 + 8x - 6

[Solution]

 Write a Quadratic Function in Standard Form
  f(x) = 2x2 - x - 6

Solve

Step

Graph

Factor out the 2 first 
Multiply out to verify

Now complete the square* inside parenthesis
To Remove -1/16 from parenthesis, Multiply it by 2.
(x - 1/4)2 = x2 - (1/2)x + 1/16

Simplify the constants

Vertex (1/4, -49/8)

Vertex is (h,k)
This means the vertex is shifted 1/4 unit right and -49/8 units down from the origin.

This graph was done with the Online Grapher.              Click here ....

*Check out  completing the square for help with this step.

 Write a Quadratic Function in Standard Form
                                 f(x) = 3x2 - x 4

[Solution]

Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada