Final answer:
The range of values for c that ensures the curve y = 3x^2 - 2x + c - 1 intersects the x-axis at two points is c > -1, as this results in a discriminant greater than zero which indicates two distinct real roots.
Step-by-step explanation:
To determine the range of values for c for which the curve y = 3x^2 - 2x + c - 1 cuts the x-axis at two points, we need to consider the discriminant of the quadratic equation. For a quadratic equation ax^2 + bx + c = 0, the discriminant is given by b^2 - 4ac. If the discriminant is greater than 0, the equation has two distinct real roots, which means the curve intersects the x-axis at two points.
In the given equation, a = 3, b = -2, and c is the constant we are trying to find. Substituting a and b into the discriminant formula gives us (-2)^2 - 4(3)(c - 1). For the curve to cut the x-axis at two points, this discriminant must be greater than 0, which leads us to the inequality 4 - 12(c - 1) > 0. Solving for c gives us c > -1.
So, the correct range of values for c such that the curve intersects the x-axis at two points is c > -1.