Denote the interpolating quadratic polynomial by p(x) = \(\displaystyle{a}{0}+{a}{1}{x}+{a}{2}{x}^{{2}}\)

Let's substitute the given points in the equation of the polynomial. The points (0,0), (—1,1) and (1,1) thus have to satisfy

0=a0

\(1=a_0-a_1+a_2\)

\(1=a_0+a_1+a_2\)

We can solve the system by inspection. Substitute ag = 0 into the second and the third equation. We get a system of two equations in two unknowns: \(-a_1+a_2=1\)

Sum the equations to get \(2a_2=2\)

This implies \(a_2 = 1\). Substituting back into the equations, we get \(a_1 =1-a_2=1-1=0\)

Therefore, the interpolating polynomial is \(\displaystyle{p}{\left({x}\right)}={x}^{{2}}\)