Question

What is the equation of the line tangent to `f(x)=-x^2 + 4x - 9` at `x=-1`?

Answer 1

`y+14 = 6(x+1)` is the equation of tangent line in the point-slope form. The step by step explanation is given below

Explanation:

To find equation of tangent we need to find two things
1) Find the slope `m`
2) A point `(x_1,y_1)`

We are asked to find at `x=-1`
`f(x)=-x^2+4x-9`

`f(-1)=-(-1)^2+4(-1)-9`
`f(-1)=-1-4-9`
`f(-1)=-14`

The point `(x_1,y_1) = (-1,-14)`

To find slope we need to find the derivative of `f(x)` at `x=-1`

`f(x) = -x^2+4x-9`
`f'(x) = -2x + 4`
`m=f'(-1) = -2(-1)+4`
`m=2+4`
`m=6`

Equation of a line passing through `(x_1,y_1)` with slope `m` is given by

`y-y_1 = m(x-x_1)`

`y-(-14) = 6(x+1)`
`y+14 = 6(x+1)` is the equation of tangent line in the point slope form.