SOLUTION: I need help with this homework question: Find the standard form of the function f(x)=x^2-4x+6. I do not understand this at all.

Algebra ->  Functions -> SOLUTION: I need help with this homework question: Find the standard form of the function f(x)=x^2-4x+6. I do not understand this at all.      Log On


   

Question 86029: I need help with this homework question: Find the standard form of the function f(x)=x^2-4x+6. I do not understand this at all.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=1+x%5E2-4+x%2B6 Start with the given equation



y-6=1+x%5E2-4+x Subtract 6 from both sides



y-6=1%28x%5E2-4x%29 Factor out the leading coefficient 1



Take half of the x coefficient -4 to get -2 (ie %281%2F2%29%28-4%29=-2).


Now square -2 to get 4 (ie %28-2%29%5E2=%28-2%29%28-2%29=4)





y-6=1%28x%5E2-4x%2B4-4%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 4 does not change the equation




y-6=1%28%28x-2%29%5E2-4%29 Now factor x%5E2-4x%2B4 to get %28x-2%29%5E2



y-6=1%28x-2%29%5E2-1%284%29 Distribute



y-6=1%28x-2%29%5E2-4 Multiply



y=1%28x-2%29%5E2-4%2B6 Now add 6 to both sides to isolate y



y=1%28x-2%29%5E2%2B2 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=1, h=2, and k=2. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=1x%5E2-4x%2B6 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1x%5E2-4x%2B6%29 Graph of y=1x%5E2-4x%2B6. Notice how the vertex is (2,2).



Notice if we graph the final equation y=1%28x-2%29%5E2%2B2 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1%28x-2%29%5E2%2B2%29 Graph of y=1%28x-2%29%5E2%2B2. Notice how the vertex is also (2,2).



So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.