# Tangents & Normals

In this section we learn how to find the equation of the tangent and the normal to a curve at a point along its length.

For each we learn a twp-step method as well as view a tutorial and work our way through exercises to consolidate our knowlede.

The following graph illustrates the tangent and the normal to the curve $$y=x^2-1$$ at the point $$\begin{pmatrix}2,3\end{pmatrix}$$:

## Method : Tangent to a Curve

Given a function $$f(x)$$, described by a curve $$y=f(x)$$, we find the equation of the tangent to the curve at a point $$\begin{pmatrix}a,b\end{pmatrix}$$ along its length, in two steps:

• Step 1: find the gradient $$m$$.

The tangent's gradient equals to the curve's gradient at the point $$\begin{pmatrix}a,b\end{pmatrix}$$, which equals to the derivative $$f'(x)$$ evaluated at $$x=a$$: $m = f'(a)$
• Step 2: find the tangent's equation, $$y=mx+c$$, by making $$y$$ the subject in the formula: $y-b = m \begin{pmatrix}x - a \end{pmatrix}$

## Example

Find the equation of the tangent to the curve $y = \frac{x^2}{2}+x + \frac{3}{2}$ at the point along its length with coordinates $$\begin{pmatrix}1,3\end{pmatrix}$$.