Final answer:
To find the second intersection point of the normal to the parabola at a given point, calculate the slope of the tangent, find the negative reciprocal for the normal's slope, and solve a quadratic equation to find the other point of intersection.
Step-by-step explanation:
The question inquires about the intersection point of the normal to a parabola, y = 3x - 3 + 4x², at a given point (1, 4), with the parabola itself at a second point. To find this second intersection point, one would first determine the derivative of the parabola at (1, 4) to find the slope of the tangent at that point. Then, by finding the negative reciprocal of this slope, one can determine the slope of the normal line. After obtaining the equation of the normal line, one would set this equation equal to the original parabola's equation and solve for x, excluding the x-value of 1 as we are looking for the other intersection point. This process involves finding the roots of a quadratic equation. The two potential lines mentioned, Y2 and Y3, are not directly related to this question as they involve lines of best fit rather than a geometric normal to a curve.
For any algebraic calculations such as finding the intersection, one may use the quadratic equation formula to solve for the roots when a quadratic equation is given in the form at² + bt + c = 0.