1. **Problem Statement:** Understand how to graph polynomials in Grade 12 Advanced Functions. 2. **Key Concepts:** A polynomial function is of the form $$f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$$ where $n$ is a non-negative integer and $a_n \neq 0$. 3. **Degree and Leading Coefficient:** The degree $n$ determines the general shape and number of turning points (maximum $n-1$). 4. **End Behavior:** Determined by the leading term $a_nx^n$: - If $n$ is even and $a_n > 0$, both ends go up. - If $n$ is even and $a_n < 0$, both ends go down. - If $n$ is odd and $a_n > 0$, left end down, right end up. - If $n$ is odd and $a_n < 0$, left end up, right end down. 5. **Intercepts:** - **Y-intercept:** $f(0) = a_0$. - **X-intercepts (roots):** Solve $f(x) = 0$. 6. **Multiplicity of Roots:** - If a root has even multiplicity, the graph touches the x-axis and turns around. - If odd multiplicity, the graph crosses the x-axis. 7. **Turning Points:** Maximum of $n-1$ turning points. 8. **Graphing Steps:** - Find intercepts. - Determine end behavior. - Identify multiplicity of roots. - Plot points and sketch smooth curve. 9. **Example:** Graph $f(x) = x^3 - 3x^2 + 2$. - Find roots: $x^3 - 3x^2 + 2 = 0$. - Use Rational Root Theorem or factorization. 10. **Summary:** Understanding degree, leading coefficient, roots, multiplicity, and intercepts is essential for graphing polynomials.