Question

Complete parts (a)-(c)a) Find a cubic function that models the data in the table. Report the model with three decimal places.

Answer 1

Step-by-step explanation:

At x = 3, we have y = 0

This means x = 3 is a root.

Let the cubic function be:

`y=(x-3)(x-a)(x-b)_{}`

When x = 1, y = -2, so

`\begin{gathered} -2=(1-3)(1-a)(1-b) \\ 1=(1-a)(1-b) \\ 1=1-b-a+ab \\ a+b-ab=0\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots..........\ldots\ldots\text{.}(1) \end{gathered}`

SImilarly, when x = 2, y = -1

`\begin{gathered} -1=(-1-3)(-1-a)(-1-b) \\ -1=4(1+a)(1+b)_{} \\ (1+a)(1+b)=-(1)/(4) \\ \\ 1+b+a+ab=-(1)/(4) \\ \\ a+b+ab=-(5)/(4)\ldots\ldots\ldots.....\ldots\ldots..\ldots.\ldots\ldots\text{.}(2) \end{gathered}`

Adding (1) and (2)

`\begin{gathered} 2a+2b=-(5)/(4) \\ \\ a+b=-(5)/(8)\ldots\ldots\ldots\ldots\ldots\ldots...........\ldots\ldots\ldots\ldots(3) \end{gathered}`

Subtracting (1) from (2)

`\begin{gathered} 2ab=-(5)/(4) \\ \\ ab=-(5)/(8)\ldots\ldots..\ldots.\ldots\ldots\ldots\ldots\ldots\ldots\ldots\text{.}(4) \\ \\ \Rightarrow b=-(5)/(8a)\ldots\ldots\ldots\ldots\ldots\ldots\ldots.\ldots\ldots.\text{.}(5) \end{gathered}`

Using (5) in (3)

`\begin{gathered} a-(5)/(8a)=-(5)/(8) \\ \\ 8a^2-5=-5a \\ 8a^2+5a-5=0 \\ a=(-5)/(16)-\sqrt[]{(185)/(16)} \\ \\ OR \\ (-5)/(16)+\sqrt[]{(185)/(16)} \end{gathered}`

Therefore, for b, we have:

`\begin{gathered} b=(1)/(16)(√(185)-5) \\ OR \\ b=(1)/(16)(-5-\sqrt[]{185}) \end{gathered}`

Replacing a and b by these values obtained, we have the required cubic function