Question

How do you find the equation of a line tangent to the function `y=2x^2-5x` at (2,-2)?

Answer 1

y = 3x - 8

Explanation:

The equation of the tangent line in `color(blue)"point-slope form"` is

`color(red)(|bar(ul(color(white)(a/a)color(black)(y-y_1=m(x-x_1)color(white)(a/a)|)))`
where m represents the gradient and `(x_1,y_1)" a point on the tangent"`

The gradient (m) of the tangent is the value of `dy/dx` at x = 2

`rArrdy/dx=4x-5" and at x = 2", dy/dx=4(2)-5=3`

We now have what is required to obtain the equation.

That is m = 3 and `(x_1,y_1)=(2,-2)`

Substitute these values into the point-slope form.

`y-(-2)=3(x-2) rArry+2=3x-6`

`rArry=3x-8" is the equation of the tangent line"`