Where does the normal line to the parabola, given below, at the given point, intersect the parabola a second time? illustrate with a sketch. (round the answers to three decimal places.) y = 4 x - 2 x^ 2 p = (2, 0)?

The graph of the parabola when sketched is shown in the picture illustrated as the blue curve. If you want to find the normal line to the parabola at point (2,0), you want to find the line perpendicular to the curve at that certain point. Note that two equations are perpendicular to each other when the product of their slopes is equal to -1. With that, let's find the slope of the parabola at point (2,0). The slope can be determined by finding the first derivative of the equation and substituting the x-coordinate of the point.

y' = 4 - 4x = 4-4(2) = -4
Thus, the slope of the normal line is 1/4 (negative reciprocal of -4). Its equation would be y = 1/4 x + b. To find b (y-intercept), substitute the coordinates of point (2,0):

0 = 1/4 (2) + b
b = -0.5

Therefore, the equation of the normal line is y = 1/4 x - 0.5. This is illustrated as the orange line in the picture.

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