SOLUTION: What is the vertex and axis of symmetry of the equation y=-2x^2+x+1? So far I have x=-1+or-sqrt1^2+4(-2)*1/2(-2) and have found the x intercepts to be (-1/2,1).

Algebra ->  Graphs -> SOLUTION: What is the vertex and axis of symmetry of the equation y=-2x^2+x+1? So far I have x=-1+or-sqrt1^2+4(-2)*1/2(-2) and have found the x intercepts to be (-1/2,1).      Log On


   

Question 83867: What is the vertex and axis of symmetry of the equation y=-2x^2+x+1?
So far I have x=-1+or-sqrt1^2+4(-2)*1/2(-2) and have found the x intercepts to be (-1/2,1).
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=-2+x%5E2%2B1+x%2B1 Start with the given equation



y-1=-2+x%5E2%2B1+x Subtract 1 from both sides



y-1=-2%28x%5E2%2B%28-1%2F2%29x%29 Factor out the leading coefficient -2



Take half of the x coefficient -1%2F2 to get -1%2F4 (ie %281%2F2%29%28-1%2F2%29=-1%2F4).


Now square -1%2F4 to get 1%2F16 (ie %28-1%2F4%29%5E2=%28-1%2F4%29%28-1%2F4%29=1%2F16)





y-1=-2%28x%5E2%2B%28-1%2F2%29x%2B1%2F16-1%2F16%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 1%2F16 does not change the equation




y-1=-2%28%28x-1%2F4%29%5E2-1%2F16%29 Now factor x%5E2%2B%28-1%2F2%29x%2B1%2F16 to get %28x-1%2F4%29%5E2



y-1=-2%28x-1%2F4%29%5E2%2B2%281%2F16%29 Distribute



y-1=-2%28x-1%2F4%29%5E2%2B1%2F8 Multiply



y=-2%28x-1%2F4%29%5E2%2B1%2F8%2B1 Now add 1 to both sides to isolate y



y=-2%28x-1%2F4%29%5E2%2B9%2F8 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=-2, h=1%2F4, and k=9%2F8. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=-2x%5E2%2B1x%2B1 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-2x%5E2%2B1x%2B1%29 Graph of y=-2x%5E2%2B1x%2B1. Notice how the vertex is (1%2F4,9%2F8).



Notice if we graph the final equation y=-2%28x-1%2F4%29%5E2%2B9%2F8 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-2%28x-1%2F4%29%5E2%2B9%2F8%29 Graph of y=-2%28x-1%2F4%29%5E2%2B9%2F8. Notice how the vertex is also (1%2F4,9%2F8).



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