Question

answer parts a.- e. for the function shown below. f(x)=3x^{2}+x^{3} (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 = - 3. (use a comma to separate answers as needed.) b. there are no x - intercepts at which the graph crosses the x - axis. at which x - intercepts does the graph of the function 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 graph of the function touches the x - axis and turns around at x = 0. (use a comma to separate answers as needed.) b. there are no x - intercepts at which the graph touches the x - axis and turns around. c. find the y - intercept. y = 0 d. determine whether the graph has y - axis symmetry, origin symmetry, or neither. choose the correct answer below. origin symmetry y - axis symmetry neither

Explanation:

Step1: Find x - intercepts

Set \(y = f(x)=3x^{2}+x^{3}=x^{2}(x + 3)=0\). Solving \(x^{2}(x + 3)=0\) gives \(x=0\) and \(x=-3\). The multiplicity of \(x = 0\) is 2 (even), so the graph touches the \(x\) - axis at \(x = 0\). The multiplicity of \(x=-3\) is 1 (odd), so the graph crosses the \(x\) - axis at \(x=-3\).

Step2: Find y - intercept

Set \(x = 0\) in \(y=f(x)\). Then \(y=3\times0^{2}+0^{3}=0\).

Step3: Check for symmetry

Replace \(x\) with \(-x\) in \(y = 3x^{2}+x^{3}\), we get \(y=3(-x)^{2}+(-x)^{3}=3x^{2}-x^{3}\). Since \(f(-x)
eq f(x)\) and \(f(-x)
eq - f(x)\), the graph has neither \(y\) - axis nor origin symmetry.

Answer:

a. The graph of the function crosses the \(x\) - axis at \(x=-3\).
b. The graph of the function touches the \(x\) - axis and turns around at \(x = 0\).
c. \(y = 0\)
d. neither