find the quartic function that is the best fit for the data in the table below. report the model with three…

Explanation:

Step1: Set up the general quartic function

The general form of a quartic function is $y = ax^{4}+bx^{3}+cx^{2}+dx + e$. We have 7 data - points $(x,y)$: $(-2,-9),(-1,0),(0,0),(1,0),(2,-9),(3,-54),(4,-180)$. Substitute these points into the general form to get a system of linear equations. When $x = 0,y = 0$: $0=a(0)^{4}+b(0)^{3}+c(0)^{2}+d(0)+e$, so $e = 0$. When $x=-2,y = - 9$: $-9=a(-2)^{4}+b(-2)^{3}+c(-2)^{2}+d(-2)$ $-9 = 16a-8b + 4c-2d$. When $x=-1,y = 0$: $0=a(-1)^{4}+b(-1)^{3}+c(-1)^{2}+d(-1)$ $0=a - b + c - d$. When $x = 1,y = 0$: $0=a(1)^{4}+b(1)^{3}+c(1)^{2}+d(1)$ $0=a + b + c + d$. When $x = 2,y=-9$: $-9=a(2)^{4}+b(2)^{3}+c(2)^{2}+d(2)$ $-9 = 16a+8b + 4c+2d$. When $x = 3,y=-54$: $-54=a(3)^{4}+b(3)^{3}+c(3)^{2}+d(3)$ $-54 = 81a+27b + 9c+3d$. When $x = 4,y=-180$: $-180=a(4)^{4}+b(4)^{3}+c(4)^{2}+d(4)$ $-180 = 256a+64b + 16c+4d$.

Step2: Simplify the system of equations

Add the equations $0=a - b + c - d$ and $0=a + b + c + d$: $(a - b + c - d)+(a + b + c + d)=0 + 0$ $2a + 2c=0$, so $c=-a$. Add the equations $-9 = 16a-8b + 4c-2d$ and $-9 = 16a+8b + 4c+2d$: $(-9)+(-9)=(16a-8b + 4c-2d)+(16a+8b + 4c+2d)$ $-18 = 32a+8c$. Substitute $c=-a$ into $-18 = 32a+8c$: $-18 = 32a+8(-a)$ $-18 = 32a-8a$ $-18 = 24a$, so $a=-0.75$. Since $c=-a$, then $c = 0.75$. Substitute $a=-0.75$ and $c = 0.75$ into $0=a + b + c + d$ and $0=a - b + c - d$, and then solve for $b$ and $d$. From $0=a + b + c + d$ and $0=a - b + c - d$, we can also use matrix - method or substitution method. Substitute $a=-0.75$ and $c = 0.75$ into $0=a + b + c + d$ gives $0=-0.75 + b+0.75 + d$, so $b=-d$. Substitute $a=-0.75,c = 0.75,b=-d$ into $-54 = 81a+27b + 9c+3d$: $-54=81(-0.75)+27(-d)+9(0.75)+3d$ $-54=-60.75-27d + 6.75+3d$ $-54=-54-24d$ $24d = 0$, so $d = 0$ and $b = 0$.

Answer:

$y=-0.75x^{4}+0.75x^{2}$