1. The problem is to create a picture using at least ten graphed relations, including at least two odd polynomial functions and two even polynomial functions, with proper domain and range labeling. 2. An odd polynomial function has the form $f(x) = a(x - h)^n + k$ where $n$ is an odd integer $\geq 3$. These functions are symmetric about the origin and typically have at least one relative minimum or maximum. 3. An even polynomial function has the form $f(x) = a(x - h)^n + k$ where $n$ is an even integer $\geq 4$. These functions are symmetric about the y-axis and often have a relative minimum or maximum at the vertex. 4. To create the butterfly shape, you can combine multiple polynomial functions, lines, and other relations. For example, use odd polynomials for the upper wings and even polynomials for the lower wings. 5. Example odd polynomial function: $$f_1(x) = (x - 1)^3 - 2$$ with domain $[-2, 3]$ and range $[-10, 5]$. 6. Example even polynomial function: $$f_2(x) = -0.5(x + 1)^4 + 3$$ with domain $[-3, 1]$ and range $[-5, 3]$. 7. Label each function on the graph with its number and state the domain and range clearly. 8. For the selected odd polynomial $f_1(x)$: - Relative minimum at approximately $x=1$ with value $-2$. - Domain is $[-2, 3]$. - Range is $[-10, 5]$. - Increasing on $[1, 3]$, decreasing on $[-2, 1]$. 9. For the selected even polynomial $f_2(x)$: - Relative maximum at $x=-1$ with value $3$. - Domain is $[-3, 1]$. - Range is $[-5, 3]$. - Decreasing on $[-3, -1]$, increasing on $[-1, 1]$. 10. Use colors to differentiate the functions and make the butterfly visually appealing. This approach satisfies the project requirements and creates a recognizable butterfly shape using polynomial functions.