Tangents and normals are lines at a given point on a curve. A tangent runs parallel with the curve at the point and the normal is perpendicular to the curve.

## Tangents

Recall that the equation of a straight line is given by $y=mx+c$ . $m$ is the gradient and $c$ is the $y$-intercept. Since tangents are straight lines, they will also take this form. The TANGENT to a curve at a given point has the same gradient as the curve at that point. Finally, the $y$-intercept can be found using the coordinates of the point.

ExampleFind the equation of the tangent to $y=x^2$ at the point (3,9).

## Normals

The NORMAL to a point on a curve is the line that intersects the curve at that point. Specifically, it is perpendicular to the curve at that point. It follows that the normal is perpendicular to the tangent at the given point. If the tangent has a gradient of $m$, the normal has a gradient of $-\frac{1}{m}$. The $y$-intercept of the normal, which is different from that of the tangent, can also be found using the coordinates of the given point.

ExampleFind the equation of the normal to $y=x^2$ at the point (3,9).