1. **Problem Statement:** Design a roller coaster model using polynomial functions, including at least one cubic and one quartic polynomial, and demonstrate understanding of polynomial operations, factorization, transformations, graphing, and analysis. 2. **Choosing Polynomials:** - Select a cubic polynomial, e.g., $f(x) = ax^3 + bx^2 + cx + d$ with degree 3. - Select a quartic polynomial, e.g., $g(x) = ex^4 + fx^3 + gx^2 + hx + i$ with degree 4. - Identify degree by highest power of $x$ or counting turning points. 3. **Factorization and Expansion:** - Write polynomials in factorized form, e.g., $f(x) = (x - r_1)(x - r_2)(x - r_3)$. - Expand using distributive property or binomial theorem. - Perform addition, subtraction, multiplication of polynomials. - Use long or synthetic division to divide polynomials. - Apply remainder theorem: if $f(r) = 0$, then $(x-r)$ is a factor. - Fully factorize to find all $x$-intercepts. - Consider piecewise polynomials for different track sections. 4. **Shifting and Transformations:** - Shift polynomial vertically by adding/subtracting constant $k$: $f(x) + k$. - Shifts affect maxima/minima by moving them up/down. - $x$-intercepts remain unchanged; $y$-intercept shifts by $k$. - Overall shape remains but position changes. 5. **Graphing:** - Sketch graphs labeling $x$-intercepts (roots), $y$-intercept, relative maxima and minima. - Mark intervals where function increases or decreases. - Analyze end behavior using leading coefficient test: - For cubic $ax^3$, if $a>0$, $f(x) o - o $ as $x o - o -$ and $f(x) o o $ as $x o o $. - For quartic $ex^4$, if $e>0$, $f(x) o o $ as $x o o $ and $f(x) o o $ as $x o - o -$. 6. **Verification and Analysis:** - Confirm polynomial matches thread path. - Analyze domain (all real numbers) and range (based on minima/maxima). - Discuss degree and leading coefficient impact on shape and end behavior. - Reflect on design meeting roller coaster requirements. 7. **Optional Extensions:** - Investigate complex roots using quadratic formula or polynomial division. - Calculate sum/product of roots using coefficients. - Explore slant asymptotes if rational functions are used. **Final Answer:** A roller coaster model can be represented by a cubic polynomial such as $$f(x) = (x-1)(x+2)(x-3) = x^3 - 2x^2 - 5x + 6$$ and a quartic polynomial such as $$g(x) = (x-1)(x+1)(x-2)(x+2) = x^4 - 5x^2 + 4$$ shifted vertically by adding 3 to ensure positive $y$-values: $$g(x) + 3 = x^4 - 5x^2 + 7$$. These polynomials can be graphed, analyzed for intercepts, maxima, minima, and end behavior to model a roller coaster track.