Solving X-Intercepts of a Cubic Function
The image displays a math question with two sub-questions.
Part a) is already solved in the image, where the given cubic function (x^3 - 6x^2 + 11x - 6) is factorized into (x - 1)(x - 2)(x - 3).
For part b), we need to determine the x-intercepts of the cubic function. The x-intercepts of a function occur where the function equals zero. Since we have the factorized form of the cubic function, the x-intercepts can be found by setting each factor equal to zero and solving for x.
Setting the factors equal to zero gives us:
1. x - 1 = 0, solving for x gives x = 1
2. x - 2 = 0, solving for x gives x = 2
3. x - 3 = 0, solving for x gives x = 3
Therefore, the x-intercepts are x = 1, x = 2, and x = 3.
To sketch the graph, you would plot these x-intercepts on a coordinate axis. The cubic function is positive to the right of x = 3, negative between x = 2 and x = 3, positive between x = 1 and x = 2, and negative to the left of x = 1. The overall shape of the graph is an S-curve, with the ends of the curve heading upwards to the right and downwards to the left, considering it is a positive cubic polynomial. The turning points of the function will be between the x-intercepts, but without additional information, we cannot accurately determine their locations—just that they will be somewhere between the x-intercepts we have found.
