Question

Please help me help help me please help help me out day

Answer 1

Answer:

The equation is:

`y=-2x^3+2x^2-4x+3`

Step-by-step explanation:

Let the equation be:

`y=ax^3+bx^2+cx+d`

Using (1, -1), x = 1, y = -1, we have:

`a+b+c+d=-1\ldots.\ldots\ldots....\ldots\ldots\ldots\ldots.......\ldots.\ldots\ldots....(1)`

Using (-1, 11), we have:

`\begin{gathered} -a+b-c+d=11 \\ a-b+c-d=-11\ldots\ldots..\ldots..\ldots\ldots..\ldots..\ldots\ldots\ldots\ldots\ldots\ldots\ldots..(2) \end{gathered}`

Add (1) and (2)

`\begin{gathered} 2a+2c=-12 \\ a+c=-6\ldots\ldots...\ldots..\ldots..\ldots..\ldots\ldots\ldots.\ldots\ldots\ldots\ldots\ldots\text{.}(3) \\ c=-6-a\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\text{.}(3i) \end{gathered}`

Substituting (3) into (2), we have:

`\begin{gathered} -6-b-d=-11 \\ -b-d=-11+6=-5 \\ b+d=5\ldots.\ldots..\ldots\ldots\ldots......\ldots\ldots\ldots\ldots\ldots\ldots.........(4) \\ d=5-b\ldots\ldots\ldots\ldots\ldots.........\ldots\ldots....\ldots\ldots....\ldots\ldots.(4i) \end{gathered}`

Using (2, -13)

`\begin{gathered} 8a+4b+2c+d=-13 \\ using\text{ (3i) and (4i)} \\ 8a+4b+2(-6-a)+(5-b)=-13 \\ 8a+4b-12-2a+5-b=-13 \\ 6a+3b=-6 \\ 2a+b=-2\ldots\ldots..\ldots..\ldots..\ldots\ldots....\ldots\ldots\ldots..\ldots.\ldots.\ldots\ldots..(5) \end{gathered}`

Using (3, -45)

`\begin{gathered} 27a+9b+3c+d=-45 \\ \text{using (3i) and (4i)} \\ 27a+9b+3(-6-a)+(5-b)=-45 \\ 27a+9b-18-3a+5-b=-45 \\ 24a+8b=-32 \\ 3a+b=-4\ldots\ldots\ldots\ldots\ldots.\ldots..\ldots\ldots....\ldots.\ldots.\ldots.\ldots..\ldots(6)_{} \end{gathered}`

Subtract (5) from (6)

`a=-2\ldots.\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots................\ldots\ldots..\ldots\ldots..\mathrm{}(A)`

Substitute (A) into (6)

`\begin{gathered} 3(-2)+b=-4 \\ -6+b=-4 \\ b=-4+6=2......\ldots\ldots....\ldots..\ldots\ldots....\ldots\ldots....\ldots\ldots....(B) \end{gathered}`

Using (3i) and (4i) with (A) and (B) respectively, we have:

`\begin{gathered} c=-6-(-2)=-4\ldots\ldots\ldots\ldots\ldots\ldots\ldots..\ldots\ldots\ldots\ldots\text{.(C)} \\ d=5-2=3\ldots\ldots\ldots\ldots\ldots......\ldots\ldots\ldots\ldots\ldots..........(D) \end{gathered}`

Using (A), (B), (C), and (D), we have our desired equation

`y=-2x^3+2x^2-4x+3`