{Use of Tech} Elliptic curves The equation y² = x³ - ax + 3, where a is a parameter, defines a well-known family of elliptic curves.


c. By experimentation, determine the approximate value of a (3 < a < 4)at which the graph separates into two curves.

Verified step by step guidance

1. Understand the problem: We are dealing with a family of elliptic curves defined by the equation y² = x³ - ax + 3, where 'a' is a parameter. Our goal is to find the approximate value of 'a' between 3 and 4 where the graph separates into two distinct curves.
2. Consider the discriminant of the cubic polynomial x³ - ax + 3. The discriminant will help us determine when the cubic has a double root, which is a key point where the nature of the graph changes.
3. Calculate the discriminant of the cubic polynomial x³ - ax + 3. The discriminant Δ of a cubic equation x³ + bx + c is given by Δ = -4b³ - 27c². In our case, b = -a and c = 3.
4. Set the discriminant equal to zero to find the critical value of 'a'. This is because a zero discriminant indicates a double root, which is where the graph changes from a single connected curve to two separate curves.
5. Solve the equation -4(-a)³ - 27(3)² = 0 for 'a'. This will give you the critical value of 'a' where the graph separates. Since we are looking for an approximate value between 3 and 4, ensure that the solution falls within this range.