Question

# Find the quadratic polynomial whose graph passes through the points (0, 0), (-1, 1), and (1, 1).

Polynomial graphs
Find the quadratic polynomial whose graph passes through the points (0, 0), (-1, 1), and (1, 1).

2021-06-14

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 $$\displaystyle{p}{\left({x}\right)}={x}^{{2}}$$