Final answer:
To find the y-intercept of the polynomial f(x), evaluate f(0). The intercepts (zeroes or roots) can be found by factoring or using numerical methods. The end behavior indicates that as x approaches positive or negative infinity, f(x) approaches infinity, due to the positive leading coefficient and highest power.
Step-by-step explanation:
The student has provided a polynomial function f(x) and would like to analyze it algebraically. There are some typos in the initial function, but I will proceed on the assumption that the correct function is f(x) = 3x4 + 3x3 − 17x2 + x − 6. Here's how to address the questions:
- a. Finding the y-intercept of f(x): The y-intercept occurs when x=0. Simply substitute x with 0 in the polynomial to get the y-coordinate of the y-intercept.
- b. Finding the intercepts (zeroes or roots) of f(x): Determine the roots of the polynomial by factoring it or using numerical methods such as synthetic division or the Rational Root Theorem.
- c. Describing the end behavior of this polynomial function: Analyze the highest power term. Since the leading coefficient (3) and the highest power (4) are positive, as x approaches infinity, f(x) will also approach infinity, and as x approaches negative infinity, f(x) will approach positive infinity as well.
To answer each part specifically:
- The y-intercept of f(x) is at (0, -6), which can be found by evaluating f(0).
- To find the zeroes of f(x), one could graph the function or find factors that satisfy f(x) = 0.
- The end behavior is such that as x → ∞, f(x) → ∞ and as x → -∞, f(x) → ∞.