Question

for the polynomial function f(x)= - 3x^4 - 9x^3, answer the parts a through e.
a. use the leading coefficient test to determine the graphs end behavior.
a. the graph of f(x) falls to the left and falls to the right.
b. the graph of f(x) falls to the left and rises to the right.
c. the graph of f(x) rises to the left and falls to the right.
d. the graph of f(x) rises to the left and rises to the right.
b. find the x - intercepts. state whether the graph crosses the x - axis, or touches the x - axis and turns around, at each intercept.
the x - intercept(s) is/are - 3,0.
(type an integer or a decimal. use a comma to separate answers as needed. type each answer only once.)
at which x - intercept(s) does the graph cross the x - axis? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the x - intercept(s) at which the graph crosses the x - axis is/are - 3,0.
(type an integer or a decimal. use a comma to separate answers as needed. type each answer only once.)
b. there are no x - intercepts at which the graph crosses the x - axis.
at which x - intercept(s) does the graph touch the x - axis and turn around? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the x - intercept(s) at which the graph touches the x - axis and turns around is/are
(type an integer or a decimal. use a comma to separate answers as needed. type each answer only once.)
b. there are no x - intercepts at which the graph touches the x - axis and turns around.
c. find the y - intercept.
the y - intercept is.
(simplify your answer. type an integer or a decimal.)

Explanation:

Step1: Determine end - behavior

For the polynomial function \(f(x)=-3x^{4}-9x^{3}\), the degree \(n = 4\) (even) and the leading coefficient \(a=-3\) (negative). According to the leading - coefficient test, when \(n\) is even and \(a<0\), as \(x\to-\infty\), \(y\to-\infty\) and as \(x\to+\infty\), \(y\to-\infty\). So the graph of \(f(x)\) falls to the left and falls to the right.

Step2: Find x - intercepts

Set \(f(x)=0\), so \(-3x^{4}-9x^{3}=0\). Factor out \(-3x^{3}\): \(-3x^{3}(x + 3)=0\). Then \(x = 0\) or \(x=-3\). The multiplicity of \(x = 0\) is 3 (odd) and the multiplicity of \(x=-3\) is 1 (odd). When the multiplicity of an \(x\) - intercept is odd, the graph crosses the \(x\) - axis at that point.

Step3: Find y - intercept

Set \(x = 0\) in the function \(f(x)=-3x^{4}-9x^{3}\). Then \(f(0)=-3(0)^{4}-9(0)^{3}=0\).

Answer:

a. A. The graph of f(x) falls to the left and falls to the right.
b. The x - intercept(s) is/are \(-3,0\). At the x - intercept(s) \(-3,0\) the graph crosses the x - axis. There are no x - intercepts at which the graph touches the x - axis and turns around.
c. The y - intercept is \(0\).