Note that, the iso$-$parametric shape functions are obtained using Lagrange Multipliers. For an $N^{th}$ degree polynomial approximation, there should be $N+1$ shape functions. Note that, the shape functions should have a value unity $($e$.$g$.,1)$ at the node that they belong to and zero $($e$.$g$.0)$ at the other nodes. For example, a first$-$order approximation would be given as:

$)=(1)=(2$

where $)=(1$ and $)=(2$ represents the nodal values of the function $u$ at the first and second nodes. Here, ${}_1(x)=\frac{x-x_2}{x_1-x_2}$ and ${}_2(x)=\frac{x-x_1}{x_2-x_1}.$

a$)$ If the element extends between coordinates $-1x+1,$ what are the shape functions in simplified forms $($e$.$g$.,x_1=-1$ and $x_2=+1)?$

b$)$ Show that ${}_1(-1)=1$ and ${}_1(+1)=0.$ Also show that ${}_2(-1)=0$ and ${}_2(+1)=1$

c$)$ If the approximation is extended to second order $($quadratic$)$ polynomials as

$)=(1)=(2)=(3$

Show that with above discussion selection of the base functions as:

${}_1(x)=\frac{x-x_2}{x_1-x_2}\frac{x-x_3}{x_1-x_3}$;${}_2(x)=\frac{x-x_1}{x_2-x_1}\frac{x-x_3}{x_2-x_3}$ and ${}_3(x)=\frac{x-x_1}{x_3-x_1}\frac{x-x_2}{x_3-x_2}$

would be a proper choice $($e$.$g$.,{}_1(x_1)=1$ and ${}_1(x_2)={}_1(x_3)=0$ and for the others, the similar rule applies$).$

d$)$ For the quadratic approximation given above and with node locations $x_1=-1,x_2=0$ and $x_3=+1$ find explicit functions for ${}_1,{}_2$ and ${}_3$