Final answer:
To sketch the graph of the polynomial function f(x) = 4x² - 2x - 6, we first factor to find the zeros, then use a graphing calculator to determine the max or min values, and finally determine the end behavior to fill in the arrows.
Step-by-step explanation:
The task is to graph the given polynomial function f(x) = -x + 4x² - x - 6. The correct simplification of this function is f(x) = 4x² - 2x - 6. First, we factor the equation to find the zeros. By using the factor theorem or a tool like synthetic division, we can look for values that make the polynomial equal to zero.
Once the zeros are found, we can use a graphing calculator to find the maximum or minimum values and determine the end behavior of the polynomial. The end behavior refers to how the function behaves as x approaches positive or negative infinity and is dependent on the leading term's degree and coefficient.
An even-degree polynomial with a positive leading coefficient will eventually rise to positive infinity on both ends (even +), while an odd-degree polynomial with a positive leading coefficient will rise to positive infinity as x goes to infinity and fall to negative infinity as x goes to negative infinity (odd +).
To complete the graph, we sketch the curve based on the zeros, maximum/minimum points, and end behavior. We fill in the arrows at the ends of the graph according to the end behavior.