SOLUTION: Find the axis of symmetry. y = –x^2 + 4x – 6

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Question 88269: Find the axis of symmetry.
y = –x^2 + 4x – 6
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%2B4+x-6 Start with the given equation



y%2B6=-1+x%5E2%2B4+x Add 6 to both sides



y%2B6=-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%2B6=-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%2B6=-1%28%28x-2%29%5E2-4%29 Now factor x%5E2-4x%2B4 to get %28x-2%29%5E2



y%2B6=-1%28x-2%29%5E2%2B1%284%29 Distribute



y%2B6=-1%28x-2%29%5E2%2B4 Multiply



y=-1%28x-2%29%5E2%2B4-6 Now add %2B6 to both sides to isolate y



y=-1%28x-2%29%5E2-2 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%2B4x-6 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-1x%5E2%2B4x-6%29 Graph of y=-1x%5E2%2B4x-6. Notice how the vertex is (2,-2).



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


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-1%28x-2%29%5E2-2%29 Graph of y=-1%28x-2%29%5E2-2. 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.