To graph the function \( g(t) = 3(t + 1)^2 + 1 \), let's break down the function and its key characteristics:

1. **Basic Form**: The function is a quadratic function, which generally has the form \( g(t) = a(t - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola, and \( a \) determines the width and direction of the parabola.

2. **Vertex**: The function is given as \( g(t) = 3(t + 1)^2 + 1 \). This can be rewritten in the standard form:
   \[
   g(t) = 3(t - (-1))^2 + 1
   \]
   Here, \( h = -1 \) and \( k = 1 \), so the vertex of the parabola is at \( (-1, 1) \).

3. **Direction**: Since the coefficient of \( (t + 1)^2 \) is positive (\( 3 \)), the parabola opens upwards.

4. **Stretch**: The coefficient \( 3 \) means the parabola is vertically stretched, making it narrower than the standard parabola \( t^2 \).

5. **Graphing Steps**:
   - Plot the vertex \( (-1, 1) \).
   - Since the parabola opens upwards and is narrower than the standard parabola, plot points on either side of the vertex and use symmetry to reflect them across the vertex.
   - Calculate additional points by substituting values of \( t \) into \( g(t) \). For example:
     - For \( t = 0 \), \( g(0) = 3(0 + 1)^2 + 1 = 4 \).
     - For \( t = -2 \), \( g(-2) = 3(-2 + 1)^2 + 1 = 4 \).
     - For \( t = 1 \), \( g(1) = 3(1 + 1)^2 + 1 = 13 \).

Here is the approximate shape of the graph:

- **Vertex** at \( (-1, 1) \).
- **Point (0, 4)** on the right and **point (-2, 4)** on the left.
- Parabola opens upwards and is narrower than \( t^2 \).

Would you like me to generate the graph or provide more details on a specific part of this process?

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**Related Questions:**
1. How would the graph change if the coefficient of \( (t+1)^2 \) were negative?
2. What is the impact of increasing the constant term \( +1 \) in the equation?
3. How does the graph compare to \( g(t) = t^2 \)?
4. How can we determine the axis of symmetry for this function?
5. What happens to the vertex if we modify the equation to \( g(t) = 3(t - 2)^2 + 1 \)?

**Tip**: Remember that the vertex form of a quadratic function directly reveals the vertex, making it easier to graph by hand.

Learn how to graph the quadratic function g(t) = 3(t + 1)^2 + 1 step-by-step. Understand the vertex form of a quadratic function, characteristics of the parabola, and graphing techniques. Suitable for high school students.

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