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Solve the system graphically. Verify the solution by substitution.

  • \(y = (x - 1)^2 + 1 \)
  • \(y =\frac{1}{3}(x - 2)^2 + 2\)



The points of intersection are \((-1,5)\) and \((2,2)\), so these are the solutions to the system.

Verify for \((-1, 5)\).

\(y = (x - 1)^2 + 1\)
Left Side Right Side
\[\begin{array}{r}
 y \\
 5 \end{array}\]
 \[\begin{array}{l}
   \left( {x - 1} \right)^2 + 1 \\
   \left( {-1 - 1} \right)^2 + 1 \\
 5  \end{array}\]
             LS = RS
\(y = \frac{1}{3} (x - 2)^2 + 2\)
Left Side Right Side
\[\begin{array}{r}
 y \\
 5 \end{array}\] 		\[\begin{array}{l}
 \frac{1}{3}\left( {x - 2} \right)^2 + 2 \\
 \frac{1}{3}\left( { - 1 - 2} \right)^2 + 2 \\
 5 \end{array}\]
             LS = RS

Verify for \((2, 2)\).

\(y = (x - 1)^2 + 1\)
Left Side Right Side
\[\begin{array}{r}
 y \\
 2 \end{array}\]
 \[\begin{array}{l}
   \left( {x - 1} \right)^2 + 1 \\
   \left( {2 - 1} \right)^2 + 1 \\
 2  \end{array}\]
             LS = RS
\(y = \frac{1}{3} (x - 2)^2 + 2\)
Left Side Right Side
\[\begin{array}{r}
 y \\
 2 \end{array}\] 		\[\begin{array}{l}
 \frac{1}{3}\left( {x - 2} \right)^2 + 2 \\
 \frac{1}{3}\left( { 2 - 2} \right)^2 + 2 \\
 2 \end{array}\]
             LS = RS