Question

Hello! I need some help with this homework question, please? The question is posted in the image below. Q8

Answer 1

The function is given to be:

$f\mleft(x\mright)=-9\mleft(x-8\mright)\mleft(x+1\mright)^2$

QUESTION A

The zeroes of the function are gotten when the function is equal to 0:

$f(x)=0$

Therefore, we can calculate the zeroes when:

$-9(x-8)(x+1)^2=0$

Dividing both sides by -9, we have:

$(x-8)(x+1)^2=0$

Recall the Zero-Product Principle:

$\begin{gathered} ab=0 \\ \text{then} \\ a=0,b=0 \end{gathered}$

Therefore, we have that:

$\begin{gathered} x-8=0 \\ x=8 \end{gathered}$

and

$\begin{gathered} x+1=0 \\ x=-1 \end{gathered}$

The zeroes are -1, 8.

The multiplicity of -1 is 2 and the multiplicity of 8 is 1.

QUESTION B

The graph of the function is shown below:

On observation of the graph, we can see that the graph touches the x-axis at x = -1, and it crosses the x-axis at x = 8.

Therefore, the graph crosses the x-axis at the larger x-intercept and the graph touches the x-intercept at the smaller x-intercept.

QUESTION C

The maximum number of turning points of a function is seen from the graph.

The maximum number of turning points on the graph is 2.