Given three points on a parabola, \(P(0, 0)\), \(Q(1, 1)\), and \(R(2, 4)\), find the equation of the parabola.

Step 1 ：Step 1: A standard form of the equation of a parabola is \(y = ax^2 + bx + c\). We can substitute the given points into this equation to form a system of equations.

Step 2 ：Step 2: Substituting \(P(0, 0)\) into the equation, we get \(0 = a*0^2 + b*0 + c\), which simplifies to \(c = 0\).

Step 3 ：Step 3: Substituting \(Q(1, 1)\) into the equation, we get \(1 = a*1^2 + b*1 + 0\), which simplifies to \(a + b = 1\).

Step 4 ：Step 4: Substituting \(R(2, 4)\) into the equation, we get \(4 = a*2^2 + b*2 + 0\), which simplifies to \(4a + 2b = 4\).

Step 5 ：Step 5: Solving the system of equations \(\begin{cases} a + b = 1\\ 4a + 2b = 4\end{cases}\) we find that \(a = 1\) and \(b = 0\).