Find the slope of the tangent line to the parabola y = 4 x^2 + 2 x + 7 at x0 = -2

(The tangent line to a curve at a point is the line passing through that point whose slope is the same as the slope of the curve at that point.)

The equation of this tangent line can be written in the form y - y0 = m(x - x0) where y0 is: ??

the slope of the tangent line is the first derivative of the parabola which is 8x+2, so m= 8

yes, I figured out that the tangent line is: 8x + 2, which I obtained by taking the derivative of the parabola. Plugging in x0 = -2 into 8x+2 we get -14, which is the answer to the first part. I just don't know what y0 equals in y - y0 = m(x - x0). I suspect this to be correct: y - y0 = 8(x+2), but I don't know how to proceed. Can I just plug in arbitrary values of x and y to get y0?

Yes, the slope of the tangent line is m = f ' (-2) = -14 as you found. y0 is just notation representing f(x0). So y0 = f(-2) = 19 for your problem. The tangent line is then y = 19 -14(x+2). Note that I moved the y0 over to the right side of the equation (personal preference on my part....).

Ok, thanks!!! :)