A quadratic function y=f(x)  if it touches the line y=x at the point x=1 and passes through the point (-1, 0) 

$\begin{array}{l}y=f\left(x\right)=a{x}^2+bx+c\\ f^{\mathrm{\prime }}\left(x\right)=2ax+b=1\\ \text{ at }x=1\\ \Rightarrow 2a+b=1----\left(1\right)\end{array}$

$\begin{array}{l}\mathrm{x}=1,\mathrm{y}=1\\ \Rightarrow f\left(1\right)=1\\ \Rightarrow a+b+c=1----\left(2\right)\\ \text{ Solving }\left(1\right)\mathrm{\&}\left(2\right)\end{array}$

$\begin{array}{l}\text{ curve passes through }(-1,0)\\ a=\frac{1}{4}\\ b=\frac{1}{2}\\ c=\frac{1}{4}\\ \mathrm{F}\left(=\frac{{\mathrm{x}}^2}{4}+\frac{\mathrm{x}}{2}+\frac{1}{4}\right)\\ =\frac{x^2+2x+1}{4}=\frac{1}{4}(x+1{)}^2\end{array}$