1. Sketch a graph of the function \( y=(x+1)(x-4)^{3}(x-2)^{2} \quad(4 \) marks)

To sketch the graph of the function \( y = (x + 1)(x - 4)^3(x - 2)^2 \), follow these steps: ### 1. **Identify the Zeros and Their Multiplicities** - **\( x = -1 \)**: - **Multiplicity:** 1 (odd) - **Behavior:** The graph crosses the x-axis at this point. - **\( x = 2 \)**: - **Multiplicity:** 2 (even) - **Behavior:** The graph touches the x-axis and bounces off (does not cross). - **\( x = 4 \)**: - **Multiplicity:** 3 (odd) - **Behavior:** The graph crosses the x-axis at this point. ### 2. **Determine the End Behavior** - **Degree of the Polynomial:** \( 1 + 3 + 2 = 6 \) (even) - **Leading Coefficient:** Positive (since all factors have positive leading coefficients when expanded) - **Behavior:** - As \( x \to -\infty \), \( y \to +\infty \). - As \( x \to +\infty \), \( y \to +\infty \). ### 3. **Plot Key Points** - **Y-intercept:** - Set \( x = 0 \): \[ y = (0 + 1)(0 - 4)^3(0 - 2)^2 = 1 \times (-64) \times 4 = -256 \] - Point: \( (0, -256) \) - **X-intercepts:** - \( (-1, 0) \) - \( (2, 0) \) - \( (4, 0) \) ### 4. **Analyze the Behavior Around Zeros** - **At \( x = -1 \):** - The graph crosses from positive to negative because the multiplicity is odd. - **At \( x = 2 \):** - The graph touches the x-axis and turns around (does not cross) because the multiplicity is even. - **At \( x = 4 \):** - The graph crosses the x-axis, similar to \( x = -1 \). ### 5. **Sketch the Graph** - **Starting from \( x \to -\infty \):** - The graph comes from \( +\infty \). - **Crosses at \( x = -1 \):** - Moves from positive to negative. - **Passes through \( (0, -256) \):** - Shows a deep dip due to the y-intercept. - **Touches and bounces at \( x = 2 \):** - The graph remains non-positive around this point. - **Crosses at \( x = 4 \):** - Moves from negative to positive. - **Ends at \( x \to +\infty \):** - The graph heads to \( +\infty \). ### 6. **Final Graph Features** - **Symmetry:** None. - **Turning Points:** At least two between the zeros due to the multiplicities. - **Shape:** Based on the multiplicities and end behavior, the graph will have a "W" shape with specific behaviors at each zero. ### **Graph Overview:** 1. **From left to right:** - Starts high (\( +\infty \)), decreases, crosses the x-axis at \( x = -1 \). 2. **Between \( x = -1 \) and \( x = 2 \):** - Continues decreasing to the y-intercept at \( (0, -256) \). 3. **At \( x = 2 \):** - Bounces off the x-axis without crossing. 4. **Between \( x = 2 \) and \( x = 4 \):** - Decreases to a local minimum. 5. **At \( x = 4 \):** - Crosses the x-axis and rises towards \( +\infty \). ### **Sample Sketch:** While I can't provide a visual sketch here, you can plot the points and behaviors described to create an accurate graph. Here's a rough outline to guide you: ``` y | | * | * | * | * * | | | |--------*--------*--------*-------- x -1 2 4 ``` - The asterisks (*) represent the curve passing through or bouncing at the x-intercepts. - The exact curvature near each intercept should follow the multiplicity rules explained above. **Remember:** The critical points and exact curvature can be further refined by taking derivatives, but for a sketch (especially worth 4 marks), the above analysis should suffice.