The given function is a cubic polynomial expressed as f(x) = x3 + 3x2 + x + 3. To analyze its graph, we should look at several key features of the cubic function:
- End Behavior: As x approaches infinity (x → ∞), the graph goes to infinity (f(x) → ∞), and as x approaches negative infinity (x → -∞), the graph falls to negative infinity (f(x) → -∞). This indicates that the graph will extend infinitely in the vertical direction in both positive and negative directions.
- Turning Points: A cubic function can have up to two turning points. By finding the first derivative f'(x), we can set it to zero and calculate critical points, from which we can determine local maxima and minima.
- Y-Intercept: To find the y-intercept, set x = 0: f(0) = 03 + 3(0)2 + 0 + 3 = 3. Thus, the graph crosses the y-axis at the point (0, 3).
- X-Intercepts: Setting the function equal to zero and solving for x will give us the x-intercepts of the function, indicating where the graph crosses the x-axis. These can be found either through factoring or numerical methods.
- Shape: The general shape of a cubic function is characterized by a single curve that may turn to form either an “S” shape or a “reverse S” shape depending on the sign and value of the leading coefficient.
In summary, the graph of the function x³ + 3x² + x + 3 is a continuously increasing curve that crosses the y-axis at (0, 3). Its end behavior reflects the polynomial’s degree, with both ends going towards infinity. Further analysis of the turning points will help us understand the specific peaks and troughs of the graph.