To rewrite these quadratic equations in vertex form, which is $y = a(x-h)^2 + k$, follow these steps:

1. Factorize or complete the square to rewrite the quadratic expression in the correct format.
2. Identify the vertex $(h, k)$.

47. $y = 2x^2 - 8x + 1$

1. Factor out $2$ from the first two terms: $y = 2(x^2 - 4x) + 1$

2. Complete the square inside the parentheses:

   • Take half the coefficient of $x$ ($-4$), square it ($(-4/2)^2 = 4$).
   • Add and subtract 4 inside the parentheses: $y = 2(x^2 - 4x + 4 - 4) + 1$

3. Group the perfect square trinomial and simplify: $y = 2((x - 2)^2 - 4) + 1$

4. Distribute $2$ and simplify: $y = 2(x - 2)^2 - 8 + 1$ $y = 2(x - 2)^2 - 7$

Vertex form: $y = 2(x - 2)^2 - 7$
Vertex: $(2, -7)$

---

49. $y = x^2 + 4x - 7$

1. Group $x^2 + 4x$: $y = (x^2 + 4x) - 7$

2. Complete the square:

   • Take half the coefficient of $x$ ($4$), square it ($(4/2)^2 = 4$).
   • Add and subtract $4$: $y = (x^2 + 4x + 4 - 4) - 7$

3. Group the perfect square trinomial and simplify: $y = ((x + 2)^2 - 4) - 7$

4. Simplify: $y = (x + 2)^2 - 11$

Vertex form: $y = (x + 2)^2 - 11$
Vertex: $(-2, -11)$

---

51. $y = -x^2 + 4x - 1$

1. Factor out $-1$ from the first two terms: $y = -(x^2 - 4x) - 1$

2. Complete the square:

   • Take half the coefficient of $x$ ($-4$), square it ($(-4/2)^2 = 4$).
   • Add and subtract $4$: $y = -((x^2 - 4x + 4 - 4)) - 1$

3. Group the perfect square trinomial and simplify: $y = -((x - 2)^2 - 4) - 1$

4. Distribute $-1$ and simplify: $y = -(x - 2)^2 + 4 - 1$ $y = -(x - 2)^2 + 3$

Vertex form: $y = -(x - 2)^2 + 3$
Vertex: $(2, 3)$

---

Final Results:

1. $y = 2(x - 2)^2 - 7$
2. $y = (x + 2)^2 - 11$
3. $y = -(x - 2)^2 + 3$

---

Do you want more details or have any questions? Here are 5 related questions to consider:

1. How do you find the vertex of a quadratic equation without rewriting it?
2. Can you explain why we complete the square in these problems?
3. What role does the leading coefficient ($a$) play in the vertex form?
4. How do you graph a quadratic function given in vertex form?
5. How do you convert from vertex form back to standard form?

Tip: When completing the square, always ensure the coefficient of $x^2$ is 1 by factoring if necessary.