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

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) (Show Source): You can put this solution on YOUR website!

Start with the given equation

Subtract from both sides

Factor out the leading coefficient

Take half of the x coefficient to get (ie ).

Now square to get (ie )

Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of does not change the equation

Now factor to get

Distribute

Multiply

Now add to both sides to isolate y

Combine like terms

Now the quadratic is in vertex form where , , and . 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 (,) and the parabola opens upward since

Check:

Notice if we graph the original equation we get:

Graph of . Notice how the vertex is (,).

Notice if we graph the final equation we get:

Graph of . Notice how the vertex is also (,).

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,

)
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