Question

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: Recall leading - coefficient test rules

For a polynomial function \(y = f(x)=a_nx^n+\cdots+a_0\), if \(n\) is even and \(a_n> 0\), the graph rises left and right; if \(n\) is even and \(a_n < 0\), the graph falls left and right; if \(n\) is odd and \(a_n>0\), the graph falls left and rises right; if \(n\) is odd and \(a_n < 0\), the graph rises left and falls right. Since the answer for part a is C (falls left and rises right), the polynomial is of odd - degree with a positive leading coefficient.

Step2: Analyze x - intercept behavior

If a factor of the polynomial is \((x - c)\) with multiplicity \(m\): if \(m\) is odd, the graph crosses the \(x\) - axis at \(x = c\); if \(m\) is even, the graph touches the \(x\) - axis and turns around at \(x = c\). Given \(x=-3,0\) are the \(x\) - intercepts. Without knowing the multiplicities of the factors corresponding to these roots, assume the simplest case where the factors are \((x + 3)\) and \(x\) (i.e., multiplicity 1 for each). When the multiplicity of a root is 1 (which is odd), the graph crosses the \(x\) - axis.

Answer:

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