Tangents and normals

Find the equation of the tangent (or normal) to the curve at the point where $x = 2$. Often they add "in the form $ax + by + c = 0$".

• The tangent touches the curve at the point, with the same gradient as the curve there.
• The normal goes through the same point, at right angles to the tangent.

Same five steps every time:

1. Find the point: if you only have $x$, put it into the curve's equation to get $y$.
2. Differentiate.
3. Put the $x$-value into $\dfrac{dy}{dx}$. That number is the tangent's gradient $m$.
4. Normal? Flip the gradient and change its sign: $-\dfrac{1}{m}$. This is the step people forget.
5. Put the point and the gradient into $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the point. Then rearrange into the form they ask for.

Why flip and change sign? Two lines at right angles have gradients that multiply to $-1$. So if the tangent has gradient $m$, the normal's gradient is $-\dfrac{1}{m}$, because $m \times \left(-\dfrac1m\right) = -1$.