Short Answer 
To analyze the end behavior of the polynomial function f(x) = 3x³ + 30x² + 75x, identify the leading term, which is 3x³. Since it is a cubic polynomial with a positive coefficient, the end behavior is that as x approaches positive infinity, y approaches positive infinity, and as x approaches negative infinity, y approaches negative infinity.
Step 1: Identify the Polynomial Function
To determine the end behavior of a polynomial function, first identify the function in question. In this case, the function is given as f(x) = 3x3 + 30x2 + 75x. This equation comprises different terms, where the leading term significantly influences the end behavior when x approaches positive or negative infinity.
Step 2: Analyze the Leading Term
The leading term is the one with the highest degree, which in this function is 3x3. Since it is a cubic polynomial with a positive coefficient (3), its end behavior can be deduced. For polynomials, if the leading term is of degree odd and has a positive coefficient, the following behaviors are observed:
- As x ‚ÄöUi +‚Äöau, then y ‚ÄöUi +‚Äöau.
- As x ‚ÄöUi -‚Äöau, then y ‚ÄöUi -‚Äöau.
Step 3: Confirm End Behavior Through Graph Analysis
Finally, graph the function to visually confirm the predicted end behavior. Upon plotting f(x), observe the graph’s behavior as x increases or decreases. From the graph, you will note:
- As x increases, y also increases toward infinity.
- As x decreases, y drops to negative infinity.
These confirmations solidify that the end behavior is: x ‚ÄöUi +‚Äöau, y ‚ÄöUi +‚Äöau and x ‚ÄöUi -‚Äöau, y ‚ÄöUi -‚Äöau.