SOLUTION: Complete the square to write each function in the form {{{f(x)=a(x-h)^2+k}}} {{{f(x)=3x^2+6x-2}}}

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Question 747981: Complete the square to write each function in the form f%28x%29=a%28x-h%29%5E2%2Bk
f%28x%29=3x%5E2%2B6x-2
Answer by MathLover1(20849) 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=3+x%5E2%2B6+x-2 Start with the given equation



y%2B2=3+x%5E2%2B6+x Add 2 to both sides



y%2B2=3%28x%5E2%2B2x%29 Factor out the leading coefficient 3



Take half of the x coefficient 2 to get 1 (ie %281%2F2%29%282%29=1).


Now square 1 to get 1 (ie %281%29%5E2=%281%29%281%29=1)





y%2B2=3%28x%5E2%2B2x%2B1-1%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 1 does not change the equation




y%2B2=3%28%28x%2B1%29%5E2-1%29 Now factor x%5E2%2B2x%2B1 to get %28x%2B1%29%5E2



y%2B2=3%28x%2B1%29%5E2-3%281%29 Distribute



y%2B2=3%28x%2B1%29%5E2-3 Multiply



y=3%28x%2B1%29%5E2-3-2 Now add %2B2 to both sides to isolate y



y=3%28x%2B1%29%5E2-5 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=3, h=-1, and k=-5. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=3x%5E2%2B6x-2 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C3x%5E2%2B6x-2%29 Graph of y=3x%5E2%2B6x-2. Notice how the vertex is (-1,-5).



Notice if we graph the final equation y=3%28x%2B1%29%5E2-5 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C3%28x%2B1%29%5E2-5%29 Graph of y=3%28x%2B1%29%5E2-5. Notice how the vertex is also (-1,-5).



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