Quadratic Functions f(x) = a(x  h)^{2} + k 
 Section 3.2  
Write a Quadratic Function in Standard Form f(x) = x^{2} + 6x + 8 
Solve 
Step 
Graph 

f(x) = x^{2} + 6x + 8 
Complete the square * 


f(x) = x^{2} + 6x + 9  9 + 8  Take half of 6 and square it  
f(x) = (x + 3)^{2}  1  (x + 3)^{2} = x^{2} + 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.  
Write a Quadratic Function in Standard Form f(x) = x^{2} + 4x + 6 
[Solution] 

Write a Quadratic Function in Standard Form f(x) = x^{2} + 10x  5 
Solve 
Step 
Graph 

f(x) = (x^{2}  10x)  5 
Factor out the negative sign first 


f(x) = (x^{2}  10x + 25  25)  5  Now complete the square* inside parenthesis  
Remove 25 from parenthesis  
f(x) = (x^{2}  10x + 25) + 25  5  Notice sign change  
f(x) = (x  5)^{2} + 20 
(x  5)^{2} = x^{2}  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.  
Write a Quadratic Function in Standard Form f(x) = x^{2} + 8x  6 

[Solution] 

Write a Quadratic Function in Standard Form f(x) = 2x^{2}  x  6 
Solve 
Step 
Graph 

Factor out
the 2 first 

Now complete the square* inside parenthesis  
To Remove 1/16 from parenthesis, Multiply it by 2.  
(x  1/4)^{2} = x^{2}  (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) = 3x^{2}  x  4 

[Solution] 

Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada