Question 407012
<pre><font face = "batangche" color = "indigo" size = 4><b>
{(-6,4),(-3,8),(1,-24),(2,-12),(6,-4)} 

Assume a polynomial equation of one less degree than the number 
of points given.  There are 5 points given, so we assume a 4th 
degree equation:

{{{y = Ax^4 + Bx^3 + Cx^2 + Dx + E}}}

Substituting each of the five given points into that equation:

{{{4 = A(-6)^4 + B(-6)^3 + C(-6)^2 + D(-6) + E}}}
{{{8 = A(-3)^4 + B(-3)^3 + C(-3)^2 + D(-3) + E}}}
{{{-24 = A(1)^4 + B(1)^3 + C(1)^2 + D(1) + E}}}
{{{-12 = A(2)^4 + B(2)^3 + C(2)^2 + D(2) + E) }}}
{{{-4 = A(6)^4 + B(6)^3 + C(6)^2 + D(6) + E }}}

That becomes the system:

{{{system(1296A-216B+36C-6D+E=4,
            81A-27B+9C-3D+E=8,
              A+B+C+D+E=-24,
          16A+8B+4C+2D+E=-12,
          1296A+216B+36C+6D+E=-4)}}}        

Solve that system of equations and get

{{{A=-1/9}}}, {{{B=0}}}, {{{C=43/9}}}, {{{D=-2/3}}}, {{{E=-28}}}

So the equation 

{{{y = Ax^4 + Bx^3 + Cx^2 + Dx + E }}}

becomes

{{{y = expr(-1/9)x^4 + expr(43/9)x^2 - expr(2/3)x -28}}}

and the graph of it is:

{{{drawing(5600/19,800,-7,7,-30,28, 
graph(5600/19,800,-7,7,-30,28, -x^4/9+(43/9)x^2-2x/3-28),

circle(-6,4,.1),circle(-3,8,.1),circle(1,-24,.1),circle(2,-12,.1),circle(6,-4,.1), locate(-6,4,"(-6,4)"),locate(-3,8,"(-3,8)"),locate(1,-24,"(1,-24)"),locate(2,-12,"(2,-12)"),locate(3.5,-4,"(6,-4)")  

)}}}

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Your second problem is done the same way.  The answer is

{{{y = expr(3/4)x^4 - expr(63/4)x^2 +63}}}

Edwin</pre>