The first two pictures are supposed to help explain how to do the problem, which is the third picture. I am in a linear algebra class using the program R, but I am still really confused.

Name: C 208 Linear Algebra Project I Part V: Polynomial Curve Fitting 350 Raw Data 3003rd Order Polymomial Fit 250 -9th Order Polynomial Fit 5th Order Polynomial Fit Concept Map: Consider N points given by: 200 S- ((x1,y1) (x2.ya)... (xn,yn) y We want to find the polynomial that passes through the points in S 150 100 50 Rule: When eachx, coordinate is distinct, there is precisely one p(x) that fits all n points. Stepi: Substitute each pair (xi yi) into p(x) and form the system: Conventional Form Polynomial . To calculate ao, a1, ..., a, . Need n+1 points, (xoxo)), (xi,(x),, (,(x)) Create n+l equations which can be solved for the n+1 unknowns as: Now, matrix A is known and the y-column matrix is also known>> augment matrix A >>rref > and find the solution; which will consist of the vector [ao, ai, an sfor any polynomial through given points, will have specific coefficients that pass through the points given. Name: 2o8 Linear Algebra Project I Example: P(x) is quadratic 1) Find p(x)- ao+ aix+a2 x whose graph passes through S By Hand: Set up the system: Plug in (xy) into P(x) and get system X = 1, y = 4: P(1) = ao + a1(1) + a2 (1)^2 >> 4 = ao + a1 + a2 (1,4) (2,0) (3,12) X-3. y 12: P(3)-ao+3a1+9a2 > 12 ao+3a1+9a2 2) System: Put into Ax b form, defining each matrix 4 a0 Solve for X: 12 a2 pression, xmin.xmax, # pts) Example: Practice in red, actual problem in blue [Alb]= 1114 Coding in R Directions: Solve 2 ways 1 2 4 o 1 3 9 12 Using the augmented matrix [A | b] a) Calc: b) X 41%1% b R : Using the inverse matrix, A., : this row of zeros, lets calc > work and give rref Example above: UsingAugmented on cale: [A | b] Using A inverse: F write X = [ ] solution # (ifbyrow not requested # R fills matrix by columns) # c ( ) always nx 1 column matrix A matrix (c(1,1,1, 1,2,3, 1,4,9,), ncol 3) b- c(4,0,12) x solve(A) % % b # %*% matrix multiplication #both methods work X solve (A,b) Writing and Graphing: Write P(x) 24 28 x+ 8 x and Graph P(x) To graph P(x), use curve command: curve(24-28x+8x2,-5,5, 100 #100 will plot 100 pts (see graph below) Your Problem Given p(x) = ao + aix + a2 x2 + a3 x3 passes through points (o,42), (1,0), (2, -40), (3, -72) . Fit a polynomial curve to the points. Follow the steps as outlined above