the graph y=x^3-4x^2+x+6 is stretched horizontally by a factor of 2 about the line x=4. what is the resulting equation

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To stretch the graph horizontally by a factor of 2 about the line x = 4, we need to modify the equation of the graph.

Step 1: Shift the graph to the left by subtracting the x-value of the line of reflection from the original equation.
- Since the line of reflection is x = 4, we need to subtract 4 from x.
- The equation becomes y = (x - 4)^3 - 4(x - 4)^2 + (x - 4) + 6.

Step 2: Adjust the stretching factor by multiplying the x-term by the reciprocal of the factor of stretching.
- The factor of stretching is 2, so we multiply the x-term (x - 4) by 1/2.
- The equation becomes y = (x - 4)^3 - 2(x - 4)^2 + (1/2)(x - 4) + 6.

Step 3: Simplify the equation if needed.
- Distribute the exponent of 3 to the (x - 4) term by cubing it.
- Distribute the exponent of 2 to the (x - 4) term by squaring it.
- Distribute the coefficient of 2 and 1/2 to their respective terms.
- The resulting equation is y = x^3 - 12x^2 + 48x - 64 - 2x^2 + 16x - 32 + (1/2)x - 2 + 6.
- Simplifying further, we have y = x^3 - 14x^2 + (105/2)x - 92.

Therefore, the resulting equation after stretching the graph horizontally by a factor of 2 about the line x = 4 is y = x^3 - 14x^2 + (105/2)x - 92.