Question

How do you find the equation of the line tangent to the graph of `y = x^2 - 3` at the point P(2,1)?

Answer 1

`4x-y=7`

Explanation:

The slope of a tangent to `y=x^2-3` is given by its derivative:
`color(white)("XXXX")m=(dy)/(dx) = 2x`

At `(x,y) = (2,1)` the slope becomes (substituting `2` for `x`
`color(white)("XXXX")m = 4`

The general slope point form for a line with slope `m` through a point `(hatx,haty)` is
`color(white)("XXXX")y-haty = m(x-hatx)`

Substituting `m=4`, `hatx=2`, and `haty=1`
`color(white)("XXXX")y-1 = 4(x-2)`

This could be re-written in standard form as
`color(white)("XXXX")4x-y = 7`