ANSWER:
(a)
Rises to the left and rises to the right
(b)
Zero(s) where the graph crosses the x-axis: 0, 1
Zero(s) where the graph touches, but does not cross the x-axis: -2
(c)
The y-intercept of the graph of f is 0
(d)
STEP-BY-STEP EXPLANATION:
We have the following polynomial function:
[tex]\begin{gathered} f\mleft(x\mright)=x\mleft(x-1\mright)\mleft(x+2\mright)^2 \\ f(x)=x^4+3x^3-4x \end{gathered}[/tex]
We can observe from the function that degree is even (4) and leading coefficient is positive(+1), which means that the behavior of the function would be:
Rises to the left and rises to the right
To know the zeros of the function, we must equal the function to 0, therefore it would remain:
[tex]\begin{gathered} x(x-1)(x+2)^2=0 \\ x=0 \\ x-1=0\rightarrow x=1 \\ (x+2)^2=0\rightarrow x+2=0\rightarrow x=-2 \end{gathered}[/tex]
Zero(s) where the graph crosses the x-axis: 0, 1
Zero(s) where the graph touches, but does not cross the x-axis: -2
(c)
We have that y-intercept, we calculate it when x is equal to 0, therefore:
[tex]\begin{gathered} f(x)=0\cdot(0-1)\cdot(0+2)^2 \\ f(x)=0 \\ y=0 \end{gathered}[/tex]
The y-intercept of the graph of f is 0
(d)
The resulting graph would be (obtained by means of a graphics program, for better understanding):
