Question: Find the domain and range of the graph.
Graph: a downward-opening parabola with vertex at approximately $(4, 2)$, extending downward on both sides with arrows; position_hint = center.
Domain: all real numbers
Range: $y \leq 2$
1. **State the problem:** We need to find the domain and range of a downward-opening parabola with vertex at $(4, 2)$.
2. **Recall the general form and properties:** A parabola opening downward can be represented as $$y = a(x - h)^2 + k$$ where $(h, k)$ is the vertex and $a < 0$.
3. **Domain:** Parabolas extend infinitely left and right, so the domain is all real numbers.
4. **Range:** Since the parabola opens downward and the vertex is the highest point, the range includes all $y$ values less than or equal to the vertex's $y$-coordinate.
5. **Write the domain and range:**
- Domain: $$(-\infty, \infty)$$
- Range: $$(-\infty, 2]$$
6. **Interpretation:** This means the parabola covers all $x$ values but $y$ values only up to $2$ inclusive.