Tangent line to a parabola

Text

Consider the parabola $y = \dfrac{x^2}{2} - 2x - 6$ and the point $A$ with abscissa $x=0$. Find the equation of the tangent line to the parabola passing through $A$.

Solution

The point $A$ has abscissa $0$, hence ordinate $y(0) = -6$: so $A=(0,-6)$ lies on the parabola. The derivative is $y'(x) = x - 2$, and the slope of the tangent at $A$ is $m = y'(0) = -2$. The tangent passes through $A=(0,-6)$ with slope $-2$: $y - (-6) = -2\,(x-0) \;\Rightarrow\; y = -2x - 6.$ $\boxed{y = -2x - 6}$