Finding the Equation of the Tangent Line

Consider a curve $y=f(x)$. tangent line A tangent to a curve is a straight line which touches the curve at a given point and represents the gradient of the curve at that point.
If $A$ is the point with $x$-coordinate $a$, then the gradient of the tangent line to the curve at this point is $f'(a)$. The equation of the tangent is; $$y-f(a) = f'(a)(x-a)$$

Example 1

Find the equation of the tangent line to $f(x)=x^2$ at the point where $x=3$.

Example 2

Find the equations of tangent lines to $f(x)=2x^2-8x$ at the point where the gradient is $4$?

Example 3

Find the equations of the tangents to the curve $f(x)=x^2+3x-10$ at the points where the curve cuts the $x$-axis.

Example 4

Find the equations of any horizontal tangent lines to $f(x)=2x^3-3x^2-12x+1$.