Find the graph for the function [math: y=12(x+1)(x+2)(x−4)] y= \frac{1}{2} (x+1)(x+2)(x-4) .

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We are given two forms of the same cubic polynomial: y = 12(x+1)(x+2)(x-4) and y = (1/2)(x+1)(x+2)(x-4). The graphs shown in the options all represent a cubic with roots at x = -2, -1, and 4, since those are the zeros of the factored form. When comparing the two expressions, the only difference is the overall vertical scale (the leading coefficient).

- Option a: Think about how a larger leading coefficient affects the graph. A coefficient as large as 12 would make the curve much steeper (steeper growth away from the x-axis) than a coefficient of 1/2. If this option shows a very steep cubic compared to the others, that could correspond to the 12 multiplier rather than 1/2. If the pictured graph is not as steep, this option would be incorrect. 
- Option b: If this option corresponds to a moderate steepness that matches a cubic with the given roots, it could represent the 1/2 multiplier. A smaller magnitude like 1/2 yields a less steep curve than 12, but still passes through the same intercepts at -2, -1, and 4. 
- Option c: This option would be incorrect if it shows a curvature or end behavior inconsistent with a positive leading coefficient for a cubic with those roots (for x → ∞, y → ∞ and for x → −∞, y → −∞). A sign flip or reversed end behavior would indicate a negative leading coefficient, which is not the case here. 
- Option d: Similarly, if this option depicts a graph with wrong relative steepness or wrong end behavior compared to the actual leading coefficient being positive, then it would be incorrect. If it somehow shows the same end behavior as the others but with a different orientation or intercepts that don’t align with the three zeros, it would be problematic.

Key ideas to differentiate the options:
- All valid graphs must pass through the x-intercepts at x = -2, -1, and 4.
- The end behavior must be: as x → ∞, y → ∞ and as x → −∞, y → −∞, since the leading coefficient is positive for both given expressions, and the degree is odd.
- The relative steepness of the graph reflects the magnitude of the leading coefficient (12 is much steeper than 1/2). Therefore, the option that shows the most modest steepness among the options is likely representing y = (1/2)(x+1)(x+2)(x-4), while the one that is noticeably steeper would correspond to y = 12(x+1)(x+2)(x-4).

In summary, the correct choice must depict a cubic with the three x-intercepts at -2, -1, and 4, with positive end behavior, and with moderate steepness appropriate to a leading coefficient of 1/2 rather than 12. The option that matches these criteria (intercepts, end behavior, and relative steepness) is the one that aligns with the description above for the 1/2 multiplier, while the steeper graph would correspond to the 12 multiplier. The other options would deviate from these core characteristics by showing incorrect end behavior, missing intercepts, or inappropriate steepness.

Find the graph for the function [math: y=12(x+1)(x+2)(x−4)] y= \frac{1}{2} (x+1)(x+2)(x-4) .

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