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.