Question: Consider the polynomial function given below: $f(x) = -\frac{2}{3}(x - 3)(x + 2)(x - 1)^2$ Use the colored dots and crosses in the graph to sketch the function. (NOTE: your sketch will be a zig-zaggy approximation... the true shape is curvy) Black dots control intercepts and must be exact. Orange crosses control function behavior. These will be graded leniently if your curve has the correct shape. Graph shape: a polynomial graph with x-intercepts at $(-2, 0)$, $(1, 0)$ with multiplicity 2, and $(3, 0)$; y-intercept at $(0, -2)$ is not shown as a black dot, and the curve is indicated by orange crosses mostly along the x-axis in the bottom-left to center area; position_hint=bottom-left

1. **State the problem:** We are given the polynomial function: $$f(x) = -\frac{2}{3}(x - 3)(x + 2)(x - 1)^2$$ We need to sketch its graph using the given intercepts and behavior hints. 2. **Identify intercepts and multiplicities:** - The zeros (x-intercepts) are at $x = -2$, $x = 1$, and $x = 3$. - The factor $(x - 1)^2$ indicates a zero at $x=1$ with multiplicity 2. 3. **Behavior at intercepts:** - At $x = -2$ and $x = 3$, the multiplicity is 1 (odd), so the graph crosses the x-axis. - At $x = 1$, multiplicity 2 (even), so the graph touches the x-axis and bounces off. 4. **Y-intercept:** Calculate $f(0)$: $$f(0) = -\frac{2}{3}(0 - 3)(0 + 2)(0 - 1)^2 = -\frac{2}{3}(-3)(2)(1) = -\frac{2}{3} \times (-6) = 4$$ So the y-intercept is at $(0, 4)$, not $-2$ as mentioned in the prompt, so we trust the calculation. 5. **End behavior:** - The polynomial is degree 4 (since $(x-1)^2$ is squared and there are three linear factors). - Leading coefficient is $-\frac{2}{3}$ (negative). - For even degree and negative leading coefficient, as $x \to \pm \infty$, $f(x) \to -\infty$. 6. **Sketching the graph:** - Start from bottom left (since $f(x) \to -\infty$ as $x \to -\infty$). - Cross the x-axis at $x = -2$ going upward. - Reach a local maximum between $-2$ and $1$. - Touch and bounce off the x-axis at $x = 1$. - Cross the x-axis at $x = 3$ going downward. - End going down to $-\infty$ as $x \to \infty$. 7. **Summary:** - Zeros: $-2$ (cross), $1$ (bounce), $3$ (cross). - Y-intercept: $(0,4)$. - End behavior: down on both ends. **Final answer:** The graph passes through $(-2,0)$ and $(3,0)$ crossing the axis, touches and bounces at $(1,0)$, passes through $(0,4)$, and goes down to $-\infty$ on both ends, matching the polynomial's behavior.