Answer:
To find the vertex of the parabola x^2 - 2x + y - 4 = 0, we can complete the square. Let's go step-by-step:
1. First, let's rearrange the equation by moving the constant term to the other side:
x^2 - 2x + y = 4
2. Now, we want to group the x-terms together and leave the y-term separate. To do this, we need to add a constant term to both sides of the equation. The constant term we add is half the coefficient of the x-term squared. In this case, the coefficient of the x-term is -2, so half of that is -1. We square -1 to get 1. So, we add 1 to both sides of the equation:
x^2 - 2x + 1 + y = 4 + 1
3. Now, let's factor the perfect square trinomial on the left side of the equation:
(x - 1)^2 + y = 5
4. The equation is now in the form (x - h)^2 + k = y, where (h, k) represents the coordinates of the vertex. So, we can see that the vertex of the parabola is at the point (1, 5).
Therefore, the vertex of the parabola x^2 - 2x + y - 4 = 0 is (1, 5)
Explanation:
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