# How do you find the equation of a line tangent to the function y=6-x^2 at (2,2)?

Jan 30, 2017

The equation of the line is $y = - 4 x + 10$

#### Explanation:

The function $6 - {x}^{2}$ is a polynomial, which is definitely differentiable, with $y ' = - 2 x$. That $- 2 x$ is also the slope of the line tangent to the graph of $y$ at any given point $x$.

Since we are trying to find the tangent at $\left(2 , 2\right)$, we know that the slope will be $- 2 \cdot 2 = - 4$. So, the equation of the line is:

$y = - 4 x + a$, where $a$ is the point at which the line intersects the $y$ axis. To find $a$, simply plug in the $x$ and $y$ coordinates of the point:

$2 = - 8 + a \implies a = 10$. The final equation is:

$y = - 4 x + 10$.