https://file.aiquickdraw.com/mathbot/file/17270393584804inx4p19.jpg

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.

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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.

Graph the parabola y = (x - 1)^2 - 4. Plot five points on the parabola: the vertex, two points to the left of the vertex, and two points to the right of the vertex.

Learn how to graph the quadratic function y = (x - 1)^2 - 4. This lesson covers key algebraic concepts, including identifying the vertex, plotting additional points, and drawing the parabola. Suitable for students in Grades 8-10.

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