Short Answer 
The leading term of the polynomial function f(x) = 3x⁶ is 3x⁶, indicating it has a degree of 6, which is even with a positive coefficient. Consequently, the end behavior of the graph shows that as x approaches both positive and negative infinity, y approaches positive infinity, confirming that the graph rises on both ends.
Step 1: Identify the Leading Term
To determine the end behavior of the polynomial function, start by identifying the leading term in the equation. For the function f(x) = 3x⁶ + 30x⁵ + 75x⁴, the leading term is 3x⁶. The degree of this polynomial is 6, which is even, and the coefficient is positive. This initial analysis is crucial for predicting how the graph behaves at the extremes.
Step 2: Analyze the End Behavior
Next, analyze the end behavior of the polynomial based on the leading term. Since the degree is even and the leading coefficient is positive, the end behavior can be summarized as follows:
- As x -> +‚àû, y -> +‚àû
- As x -> -‚àû, y -> +‚àû
This indicates that the graph rises on both ends, which reflects the nature of even-degree polynomials with positive coefficients.
Step 3: Confirm with a Graph
To visualize and further confirm your findings, plot the graph of the polynomial function using graphing software or a graphing calculator. Look out for the following key observations:
- As x increases (moving right), y values continue to increase.
- As x decreases (moving left), y values also increase.
This reinforces that for the polynomial f(x) = 3x⁶ + 30x⁵ + 75x⁴, the end behavior indicates both ends of the graph rising to infinity.