To find the vertex of the quadratic function with $x$-intercepts at $(-3, 0)$ and $(5, 0)$, we can use the fact that the vertex lies on the axis of symmetry of the parabola, which is exactly halfway between the intercepts.

Step 1: Find the midpoint between the intercepts

The $x$-coordinate of the vertex (axis of symmetry) is the midpoint of $-3$ and $5$: $x = \frac{-3 + 5}{2} = \frac{2}{2} = 1$

Step 2: Determine the $y$-coordinate of the vertex

Since the range of the function is $[-3, \infty)$, this suggests that the vertex is the minimum point, and the $y$-coordinate of the vertex is $-3$.

Conclusion

The vertex of the quadratic function is at: $(1, -3)$

Would you like a more detailed explanation of how this was determined?

Further Questions

1. How can you find the equation of a quadratic function given the intercepts and vertex?
2. What does the range tell us about the orientation of the parabola?
3. How does the vertex form of a quadratic equation relate to the function's graph?
4. How would you find the axis of symmetry in a quadratic function?
5. What are other ways to determine the vertex of a quadratic equation?

Tip

For any quadratic function, the vertex lies exactly at the midpoint of the x-intercepts if they are given.