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The question asks to graph the quadratic function \( h(x) = x^2 + 3 \), state the vertex, and give the equation for the axis of symmetry.

### Step-by-Step Solution

1. **Identify the Function**: The function \( h(x) = x^2 + 3 \) is a quadratic function in the form \( y = ax^2 + bx + c \). In this case, \( a = 1 \), \( b = 0 \), and \( c = 3 \).

2. **Determine the Vertex**: For a quadratic function in the form \( y = ax^2 + bx + c \), the vertex is at the point \((x, y)\) where \( x = -\frac{b}{2a} \).
   - Here, \( b = 0 \), so \( x = -\frac{0}{2 \times 1} = 0 \).
   - Substitute \( x = 0 \) back into the function to find \( y \):
     \[
     h(0) = (0)^2 + 3 = 3
     \]
   - Therefore, the vertex is at \( (0, 3) \).

3. **Axis of Symmetry**: The axis of symmetry for a parabola in the form \( y = ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \).
   - Since \( x = 0 \) for this function, the axis of symmetry is \( x = 0 \).

4. **Graphing the Function**:
   - Plot the vertex at \( (0, 3) \).
   - Since \( a = 1 \) (positive), the parabola opens upward.
   - For additional points, you could calculate \( h(1) = 4 \), \( h(-1) = 4 \), \( h(2) = 7 \), and \( h(-2) = 7 \).
   - Draw a solid curve through these points, making sure the parabola is symmetric about the line \( x = 0 \).

5. **Graphing the Axis of Symmetry**:
   - Draw a dashed vertical line at \( x = 0 \) to represent the axis of symmetry.

### Summary
- **Vertex**: \( (0, 3) \)
- **Axis of Symmetry**: \( x = 0 \)
- **Graph**: Parabola opening upwards, centered at \( (0, 3) \), symmetric about the line \( x = 0 \).

Would you like further details or have any questions about this solution?

### Follow-up Questions
1. How would the graph change if the function was \( h(x) = -x^2 + 3 \)?
2. What is the effect of changing the \( c \)-value in \( y = x^2 + c \) on the graph?
3. How would you find the vertex for a more general function like \( h(x) = ax^2 + bx + c \)?
4. What does the \( a \)-value determine in the shape of the parabola?
5. How do we know if a parabola opens upwards or downwards?

#### Tip:
For any quadratic function \( y = ax^2 + bx + c \), the sign of \( a \) tells you the direction of the parabola's opening: \( a > 0 \) opens upward, and \( a < 0 \) opens downward.

Sketch the graph of the quadratic function h(x) = x^2 + 3, state the vertex, and give the equation for the axis of symmetry.

Learn how to sketch the graph of the quadratic function h(x) = x^2 + 3, find its vertex, and determine the axis of symmetry. This problem covers algebraic concepts and graphing techniques suitable for students in Grades 9-10.

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