The equation you are working with is:

$y = (x - 1)^2 - 4$

This is the equation of a parabola in vertex form, $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Step-by-step Analysis:

1. Vertex:

   • From the equation $y = (x - 1)^2 - 4$, we see that $h = 1$ and $k = -4$.
   • Therefore, the vertex of the parabola is $(1, -4)$.

2. Finding additional points:

   • Now, let's calculate two points to the left of the vertex and two points to the right of the vertex by choosing different values for $x$ and calculating the corresponding $y$-values.

   • Left of the vertex (values smaller than 1):

     • When $x = 0$: $y = (0 - 1)^2 - 4 = 1 - 4 = -3 \quad \text{Point: } (0, -3)$
     • When $x = -1$: $y = (-1 - 1)^2 - 4 = 4 - 4 = 0 \quad \text{Point: } (-1, 0)$

   • Right of the vertex (values larger than 1):

     • When $x = 2$: $y = (2 - 1)^2 - 4 = 1 - 4 = -3 \quad \text{Point: } (2, -3)$
     • When $x = 3$: $y = (3 - 1)^2 - 4 = 4 - 4 = 0 \quad \text{Point: } (3, 0)$

Points to Plot:

• Vertex: $(1, -4)$
• Additional points: $(0, -3), (-1, 0), (2, -3), (3, 0)$

These five points should give you a clear shape of the parabola. You can now plot these points on the graph and draw a smooth curve through them to complete the parabola.

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How can I determine whether a parabola opens upward or downward?
2. What is the axis of symmetry for the given parabola?
3. How do changes in the $a$ value affect the width of the parabola?
4. Can you explain how to find the x-intercepts of the parabola?
5. What are the key differences between vertex form and standard form of a parabola?

Tip: When graphing parabolas, use symmetry to ensure the points on either side of the vertex mirror each other.