To determine the end behavior of the polynomial function f(x) = 6x³ + 3x² – 3x + 2, we first need to identify the leading term, which is the term with the highest power of x. In this case, the leading term is 6x³.
The leading coefficient is 6, and since the degree of the polynomial (which is 3) is odd and the leading coefficient is positive, we can use the Leading Coefficient Test to predict the end behavior of the polynomial:
- As x approaches positive infinity (x → ∞), f(x) approaches positive infinity (f(x) → ∞).
- As x approaches negative infinity (x → -∞), f(x) approaches negative infinity (f(x) → -∞).
Now that we have established the end behavior, we can summarize:
- As x goes to infinity, f(x) goes to infinity.
- As x goes to negative infinity, f(x) goes to negative infinity.
Next, to visualize this function, you would graph the polynomial. You would start by plotting points based on the function values at various x-values, particularly around the x-intercepts and y-intercept, and use the end behavior we analyzed above to ensure that the curve appropriately extends towards infinity and negative infinity at the ends. A rough sketch would show the graph starting low on the left, passing through the roots, and then continuing high on the right.
In summary, the polynomial opens upwards on the right and downwards on the left, which is characteristic of polynomials with an odd degree and a positive leading coefficient.