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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BlackTom AI · Accepted Answer

The task asks to find the graph for the function y = 12(x+1)(x+2)(x-4) and compares it to y = 1/2 (x+1)(x+2)(x-4). 
First, note that both functions share the same zeros at x = -1, x = -2, and x = 4, since the zeros come from the factors (x+1), (x+2), and (x-4). This means every candidate graph should cross the x-axis at those three x-values (assuming the graph is drawn with those intercepts clearly visible). 
Second, the two graphs have the same basic cubic shape (both have an odd degree with a positive leading coefficient). Thus, as x → ∞, y → ∞ and as x → −∞, y → −∞ for both, and the general wobble with two turning points is preserved. 
Third, the only difference between the two graphs is vertical stretch/compression due to the constant multiplier in front: 12 versus 1/2. The 12-multiplied version is 24 times taller than the 1/2 version at any nonzero x, so it will be much steeper (a larger absolute slope away from the turning points) and reach higher and lower y-values for the same x. 
To decide which option corresponds to which equation, you can use these checks:
- If you pick a vertical sample point (say x = 0) and compare the y-values for the two graphs, the 12× version will be 12 times the value of the base cubic at x = 0, while the 1/2× version will be half that. Specifically, at x = 0, y = 12(1)(2)(-4) = -96 for the 12× version, and y = 1/2(1)(2)(-4) = -4 for the 1/2× version. The graph that passes through (0, -96) is the 12× version; the one through (0, -4) is the 1/2× version. 
- If you inspect the graph for the steepness: the graph with the larger absolute leading coefficient (12) will have steeper ascent/descent away from the turning points than the one with 1/2. 
- If the option graph shows intercepts at x = -1, -2, and 4 and the curve is very tall (relative to the same axis scale) on the right side, that would indicate the y = 12(x+1)(x+2)(x-4) form; if it is more modest in height, that would indicate y = (1/2)(x+1)(x+2)(x-4).
If the provided answer choices (a, b, c, d) include graphs that fail to cross the x-axis at the required roots, or show wrong end behavior, those can be eliminated. Without seeing the actual graphs corresponding to a, b, c, or d, it isn’t possible to definitively label which option matches which equation here. 
In short, compare each option against these criteria: correct x-intercepts at -2, -1, 4; correct end behavior (upward to the right, downward to the left); and relative vertical scale (a 12× factor yields a graph that is much taller than the 1/2× version for the same x-values). If you provide the specific shapes of options a–d, I can identify exactly which one corresponds to y = 12(x+1)(x+2)(x-4) and which to y = (1/2)(x+1)(x+2)(x-4).

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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