QUIZ 3

1. Find the parabola $y=A+Bx+Cx^2$ that passes through the three points $(1,1),(2,-1)$ and $(3,1)$.
   
   • By elimination:
     
     $$\begin{array}{r} A+B+C=1\\ A+2B+4C=-1\\ A+3B+9C=1\\ \e... ...\\ (2)-(1):B+3C=-2\\ (3)-(1):2B+8C=0\\ \end{array}\mapsto$$
     
     
     
     
     $$\begin{array}{r} A+B+C=1\\ B+3C=-2\\ (3)-2*(2):2C=4\\ \end{array}$$
     
     
     The final system is upper triangular. The parabola is $y=7-8x+2x^2$
     
   • By augmented matrix
     
     
     $$\left[ \begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & 2 & 4 & -... ...1 \\ 0 & 1 & 3 & -2\\ 0 & 2 & 8 & 0\\ \end{array} \right]$$
     
     
     
     
     $$\begin{array}{r} \\ \\ R_3-(2/1)R_2 \rightarrow \\ \en... ...1 \\ 0 & 1 & 3 & -2\\ 0 & 0 & 2 & 4\\ \end{array} \right]$$
     
     
     The final system is upper triangular. The parabola is $y=7-8x+2x^2$