On a separate piece of paper, sketch the graph of the parabola $\displaystyle {y}={x}^{{2}}+{7}$. On the same graph, plot the point $\displaystyle {\left({0},-{5}\right)}$. Note that there are two tangent lines of $\displaystyle {y}={x}^{{2}}+{7}$ that pass through the point $\displaystyle {\left({0},-{5}\right)}$.

Specifically, the tangent line of the parabola $\displaystyle {y}={x}^{{2}}+{7}$ at the point $\displaystyle {\left({a},{a}^{{2}}+{7}\right)}$ passes through the point $\displaystyle {\left({0},-{5}\right)}$ where $\displaystyle {a}>{0}$. The other tangent line that passes through the point $\displaystyle {\left({0},-{5}\right)}$ occurs at the point $\displaystyle {\left(-{a},{a}^{{2}}+{7}\right)}$.

Find the number $\displaystyle {a}$. Preview Question 6