Ellipse Domain Range 6D6A24

1. **State the problem:** Find the domain and range of the ellipse given by the equation $$\frac{(x-1)^2}{4} + \frac{(y-2)^2}{9} = 1$$ and then graph it. 2. **Recall the standard form of an ellipse:** $$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$ where $(h,k)$ is the center, $a$ is the semi-major axis length, and $b$ is the semi-minor axis length. 3. **Identify parameters:** Center: $(1,2)$ Horizontal semi-axis $a = 2$ (since $a^2=4$) Vertical semi-axis $b = 3$ (since $b^2=9$) 4. **Find the domain:** The ellipse extends horizontally from $x = h - a$ to $x = h + a$. $$\text{Domain} = [1 - 2, 1 + 2] = [-1, 3]$$ 5. **Find the range:** The ellipse extends vertically from $y = k - b$ to $y = k + b$. $$\text{Range} = [2 - 3, 2 + 3] = [-1, 5]$$ 6. **Summary:** - Domain: $[-1, 3]$ - Range: $[-1, 5]$ 7. **Graph description:** The ellipse is centered at $(1,2)$ with horizontal radius $2$ and vertical radius $3$.