1. **State the problem:** We are given two cubic polynomial functions: $$P(x) = \frac{3}{2}x^3 + 2x^2 + \frac{1}{2}x + 1$$ $$Q(x) = \frac{1}{3}x^3 + x^2 + \frac{1}{5}x - 4$$ We want to understand their shapes and behavior, possibly by analyzing and comparing them. 2. **Recall the general form and properties of cubic polynomials:** A cubic polynomial has the form: $$ax^3 + bx^2 + cx + d$$ where $a \neq 0$. - The leading term $ax^3$ dominates the end behavior. - The sign of $a$ determines whether the ends go to $+\infty$ or $-\infty$. - The polynomial can have up to two turning points (local maxima or minima). 3. **Analyze $P(x)$:** - Leading coefficient: $a = \frac{3}{2} > 0$, so as $x \to \pm \infty$, $P(x) \to \pm \infty$ respectively. - The polynomial is increasing for large positive $x$ and decreasing for large negative $x$. 4. **Analyze $Q(x)$:** - Leading coefficient: $a = \frac{1}{3} > 0$, similar end behavior as $P(x)$ but with smaller leading coefficient, so it grows slower. 5. **Compare constant terms:** - $P(0) = 1$ - $Q(0) = -4$ So $P(x)$ starts higher on the y-axis. 6. **Summary:** - Both are cubic polynomials with positive leading coefficients. - $P(x)$ grows faster for large $|x|$ due to larger leading coefficient. - $P(x)$ is shifted upward compared to $Q(x)$. This explains the shapes and relative positions of the graphs. **Final answer:** The two cubic polynomials $P(x)$ and $Q(x)$ have similar end behavior but differ in growth rate and vertical shift due to their coefficients.