Final answer:
The properties of the graph of the given polynomial functions are described in detail, including the standard form, leading term, x-intercepts and multiplicities, y-intercept, number of turning points, and possible graph with end behavior.
option f is the correct
Step-by-step explanation:
Properties of the Graphs of Polynomial Functions
a. Standard form: The standard form of a polynomial function is written in descending order of exponents. For example, the standard form of the polynomial function F(x) = x⁴ - x³ - 2x² is F(x) = x⁴ - x³ - 2x².
b. Leading term: The leading term of a polynomial function is the term with the highest exponent. In this case, the leading term is x⁴.
c. X-intercepts and its multiplicities: To find the x-intercepts, set the function equal to zero and solve for x. The multiplicities of the x-intercepts are equal to the exponents of the factors. For the function F(x) = x⁴ - x³ - 2x², there are x-intercepts at x = -1 (with a multiplicity of 2) and x = 0.
d. Y-intercept: The y-intercept is the point where the graph intersects the y-axis. To find the y-intercept, plug in x = 0 into the function. For the function F(x) = x⁴ - x³ - 2x², the y-intercept is (0, 0).
e. Number of turning points: The number of turning points is equal to the degree of the polynomial minus one. The degree of the polynomial F(x) = x⁴ - x³ - 2x² is 4, so the number of turning points is 3.
f. Possible graph with end behavior: The end behavior of the graph depends on the leading term of the polynomial. Since the leading term is x⁴, the graph will have the same end behavior as the function y = x⁴. As x approaches positive infinity, the graph increases without bound, and as x approaches negative infinity, the graph decreases without bound.