Solution assignment 06 Tangent line to graph

Calculate the equation of the tangent line at the point $P(2,5)$ of the graph of the function:

$y=-x^2+2x+5$

First we calculate the slope of the tangent line at the given point and we do this by differentiating the function:

$y'=-2x+2$

In the point $(2,5)$ the slope is (always check whether such a point lies on the graph):

$y'=-2\cdot2+2=-2$

So we have to calculate the equation of the line through $(2,5)$ with slope $m=-2$ . We find the line by using the following formula:

$y-5=-2(x-2)$

or:

$y=-2x+9$