Question

answer parts a - e for the function shown below. f(x)=3x^{2}+x^{3}
a. use the leading - coefficient test to determine the graphs end behavior. which statement describes the end behavior of f(x)?
a. the graph of f(x) rises left and falls right.
b. the graph of f(x) rises left and rises right.
c. the graph of f(x) falls left and rises right.
d. the graph of f(x) falls left and falls 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. what are the x - intercepts?
x = - 3,0 (use a comma to separate answers as needed.)
at which x - intercepts does the graph of the function cross the x - axis? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the graph of the function crosses the x - axis at x = . (use a comma to separate answers as needed.)
b. there are no x - intercepts at which the graph crosses the x - axis.

Explanation:

Step1: Identify the leading - term

The function is \(f(x)=x^{3}+3x^{2}\), the leading - term is \(x^{3}\) with leading coefficient \(a = 1>0\) and degree \(n = 3\) (odd). For a polynomial function \(y = a_{n}x^{n}+a_{n - 1}x^{n - 1}+\cdots+a_{0}\), when \(n\) is odd and \(a_{n}>0\), as \(x\to-\infty\), \(y\to-\infty\) and as \(x\to+\infty\), \(y\to+\infty\). So the graph of \(f(x)\) falls left and rises right.

Step2: Find the x - intercepts

Set \(f(x)=0\), so \(x^{3}+3x^{2}=x^{2}(x + 3)=0\). Solving \(x^{2}(x + 3)=0\) gives \(x=-3\) or \(x = 0\).
For a factor \((x - c)\) of a polynomial \(P(x)\), if the exponent of \((x - c)\) is odd, the graph crosses the \(x\) - axis at \(x = c\), and if the exponent is even, the graph touches the \(x\) - axis and turns around at \(x = c\). For the factor \(x+3\) (exponent 1, odd), the graph crosses the \(x\) - axis at \(x=-3\), and for the factor \(x^{2}\) (exponent 2, even), the graph touches the \(x\) - axis at \(x = 0\).

Answer:

a. C. The graph of \(f(x)\) falls left and rises right.
b. \(x=-3,0\); The graph of the function crosses the \(x\) - axis at \(x=-3\). A. The graph of the function crosses the \(x\) - axis at \(x=-3\).