Example 1. Find the equations of the tangent and normal to the parabola $y=x^{2}+4 x+2$ at the point $A(1,7)$.
Solution. The point $A(1,7)$ lies on the parabola. We find $y^{\prime}=2 x+4$ and $y^{\prime}(1)=6$. Therefore (see point $1^{\circ}$), the equation of the tangent line is $y=7+6(x-1)$, i.e., $y=6 x+1$, and the equation of the normal (see point $2^{\circ}$) is $y=7-\frac{1}{6}(x-1)$, i.e., $x+6 y-43=0$.