Do the following for the points 4102123241 4 1 0 2 1 2 3 2 4

Do the following for the points (?4,?1),(0,?2),(1,?2),(3,2),(4,?1) 4 1 0 2 1 2 3 2 4 1 : (If you are entering decimal approximations, enter at least five decimal places.) (a) Find the equation for the best-fitting parabola y=ax2+bx+c y a x 2 b x c for these points: (b) Find the equation for the best-fitting parabola with no constant term y=ax2+bxyax2bx for these points: (c) Find the equation for the best-fitting parabola with no linear term y=ax2+cyax2c for these points: PLEASE SHOW ALL YOUR WORK FOR EVERY STEP

Solution

A. Given y= ax^2+bx+c

substituting the given points in the parabola

(-4,-1) ---> -1 = a(16) + b(-4) + c => -1=16a -4b +c ----1(a)

(0,-2) ----> -2=c ----1(b)

(1,-2)-----> -2 = a+b+c ------1(c)

(3,2) ----> 2 = 9a + 3b + c ----1(d)

(4,-1) ----> -1 = 16a + 4b + c -----1(e)

solving these equations we get a,b,c

here we have c=-2 substitute these in the given equations

from 1(a) we get 4(4a-b) = 3 ===> 4a-b = 3/4

from 1(e) we get 4(4a+b) = 1 ===> 4a+b = 1/4

adding these two equations

8a = 3/4+1/4

8a = 1

a= 1/8

substituting this in 1(a) we get b = -1/4

substituting these 3 values in the given equation we get the equation of parabola y=x^2 / 8 - x/4 -2

b.

Given y= ax^2+bx

substituting the given points in the parabola

(-4,-1) ---> -1 = a(16) + b(-4) => -1=16a -4b ----2(a)

(1,-2)-----> -2 = a+b ------2(b)

(3,2) ----> 2 = 9a + 3b ----2(c)

(4,-1) ----> -1 = 16a + 4b -----2(d)

solving these equations we get a,b

adding 2(a) and 2(d) we get

8a = -2/4

a = -1/16

substituting this in the 2(c) we get b = 41/48

hence the equation of the parabola is y =-x^2/16 + 41/48x

c.   

Given y= ax^2+c

substituting the given points in the parabola

(-4,-1) ---> -1 = a(16) + c => -1=16a +c ----3(a)

(0,-2) ----> -2 = c ----3(b)

(1,-2)-----> -2 = a+c ------3(c)

(3,2) ----> 2 = 9a + c ----3(d)

(4,-1) ----> -1 = 16a + c -----3(e)

solving these equations we get a,c

here we have c=-2 substitute these in the given equations

from 3(d) 9a-2 = 2

9a = 4

a = 4/9

the equation for parabola is y = 4/9x^2 -2