The graph of the polynomial function x³ – 3x² + x – 3 is a cubic curve that exhibits specific characteristics typical of cubic functions.
To analyze it, we can look for key features such as the number of turning points, end behavior, and intercepts. For this function, the leading term is x³, which indicates that as x approaches positive or negative infinity, the y-values will also approach positive or negative infinity, respectively.
Finding the zeros of the function by setting it equal to zero can help locate the x-intercepts. For this polynomial, we can perform synthetic division or factorization, resulting in real roots, which determine the points where the graph crosses the x-axis.
Further analysis of the first and second derivatives allows us to identify local maxima and minima as well as points of inflection. Overall, the graph will display a characteristic ‘S’ shape typical of cubic functions, reflecting one or more changes in direction.
In summary, the graph of x³ – 3x² + x – 3 features typical cubic behavior with specific intercepts and turning points, representing an important class of polynomial functions.