Find the equation of the tangent line to y=f(x) at x=a. Graph y=f(x). f(x)= x^2 -2, a = 0. I am confused.?

Question

Find the equation of the tangent line to y=f(x) at x=a. Graph y=f(x). f(x)= x^2 -2, a = 0. I am confused.

Answer 1

If f(x) = x^2 - 2, then the derivative givs you the slope of the line tangent to the curve at any point x. So let's take a derivative:

f(x) = x^2 - 2
f'(x) = 2x

If x = a, then the slope at this point is 2a.

In order to write an equation of a line, you need a slope and a point. You already have x = a, so we can plug that in to the original equation to find what y would equal:

f(x) = x^2 - 2
f(a) = a^2 - 2

So (and this will get a bit confusing) you have a slope of 2a and a point (a, a^2 - 2). You can write the equation of the line now:

y - y1 = m(x - x1)
y - (a^2 - 2) = 2a(x - a)
y - a^2 + 2 = 2ax - 2a^2
y = 2ax - a^2 + 2