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To graph the given parabola \( y = 3(x - 2)^2 - 7 \), we need to identify five points: the vertex, two points to the left of the vertex, and two points to the right of the vertex. Here's the step-by-step process:

1. **Identify the vertex**:
   - The vertex form of a parabola is \( y = a(x-h)^2 + k \).
   - Here, \( a = 3 \), \( h = 2 \), and \( k = -7 \).
   - Therefore, the vertex is at \( (2, -7) \).

2. **Calculate points to the left and right of the vertex**:
   - Choose \( x = 1 \) and \( x = 0 \) for points to the left.
   - Choose \( x = 3 \) and \( x = 4 \) for points to the right.

3. **Calculate the y-values for these x-values**:
   - For \( x = 1 \):
     \[
     y = 3(1 - 2)^2 - 7 = 3(1)^2 - 7 = 3 - 7 = -4
     \]
     Point: \( (1, -4) \)
   - For \( x = 0 \):
     \[
     y = 3(0 - 2)^2 - 7 = 3(2)^2 - 7 = 12 - 7 = 5
     \]
     Point: \( (0, 5) \)
   - For \( x = 3 \):
     \[
     y = 3(3 - 2)^2 - 7 = 3(1)^2 - 7 = 3 - 7 = -4
     \]
     Point: \( (3, -4) \)
   - For \( x = 4 \):
     \[
     y = 3(4 - 2)^2 - 7 = 3(2)^2 - 7 = 12 - 7 = 5
     \]
     Point: \( (4, 5) \)

4. **Plot these points** on the graph:
   - \( (2, -7) \)
   - \( (1, -4) \)
   - \( (0, 5) \)
   - \( (3, -4) \)
   - \( (4, 5) \)

These points can be plotted on the graph to form the parabola. 

If you have any questions or want further details, please let me know!

Here are 8 related questions for further practice:
1. How would the graph change if the coefficient of \( (x-2)^2 \) was negative?
2. What is the axis of symmetry for the given parabola?
3. How can you determine if a parabola opens upwards or downwards?
4. What effect does changing the value of \( h \) in \( (x-h) \) have on the graph?
5. How do you find the focus and directrix of a parabola given in vertex form?
6. Can you derive the standard form of the equation from the vertex form?
7. How do you determine the x-intercepts of the given parabola?
8. What is the effect of changing the value of \( k \) in the vertex form equation on the graph?

**Tip:** When graphing quadratic functions, always start by identifying the vertex, as it helps in determining the overall shape and position of the parabola on the coordinate plane.

Learn how to graph the parabola y = 3(x - 2)^2 - 7 with a detailed step-by-step guide. This problem covers key algebraic concepts, including the vertex form of a parabola and graphing quadratic functions. Suitable for high school students.

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