SOLUTION: How do I solve for this quadratic: x^2-2x+7 so that it is in vertex form? And on finding its range?

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Question 120174: How do I solve for this quadratic: x^2-2x+7
so that it is in vertex form? And on finding its range?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

y=1+x%5E2-2+x%2B7 Start with the given equation


y-7=1+x%5E2-2+x Subtract 7 from both sides


y-7=1%28x%5E2-2x%29 Factor out the leading coefficient 1


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

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




y-7=1%28x%5E2-2x%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-7=1%28%28x-1%29%5E2-1%29 Now factor x%5E2-2x%2B1 to get %28x-1%29%5E2


y-7=1%28x-1%29%5E2-1%281%29 Distribute


y-7=1%28x-1%29%5E2-1 Multiply


y=1%28x-1%29%5E2-1%2B7 Now add 7 to both sides to isolate y


y=1%28x-1%29%5E2%2B6 Combine like terms



Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=1, h=1, and k=6. Remember (h,k) is the vertex and "a" is the stretch/compression factor. Also "a" tells us which direction the parabola opens.



So in this case the vertex is (1,6) and the parabola opens upward since a%3E0


Check:

Notice if we graph the original equation y=1x%5E2-2x%2B7 we get:

graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1x%5E2-2x%2B7%29 Graph of y=1x%5E2-2x%2B7. Notice how the vertex is (1,6).


Notice if we graph the final equation y=1%28x-1%29%5E2%2B6 we get:

graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1%28x-1%29%5E2%2B6%29 Graph of y=1%28x-1%29%5E2%2B6. Notice how the vertex is also (1,6).


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




Finding the Range:

Notice how the lowest point coincides with the vertex. So the lowest point is at (1,6) which means y will never be less than 6. So the range is: y can be any number greater than or equal to 6. The range looks like this in interval notation: [6,)