Answer
(a) $d(x)= \sqrt {x^4-15x^2+64}$
(b) $8$
(c) $5\sqrt {2}$
(d) See graph below.
(e) $x\approx\pm2.74$
Work Step by Step
(a) The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the formula $d=\sqrt{(x-1-x_2)^2+(y_1-y_2)^2}$
Given $y=x^2-8$, the distance from P$(x,y)$ to the origin $(0,0)$ is:
$$\begin{align*}
d&=\sqrt {x^2+y^2}\\
d&=\sqrt {x^2+(x^2-8)^2}\\
d&=\sqrt{x^2+x^4-16x^2+64}\\
d&=\sqrt {x^4-15x^2+64}
\end{align*}$$
Thus, the distance $d$ from $P$ to the origin as a function of $x$ is: $d(x)=\sqrt{x^4-15x^2+64}$.
(b) For $x=0$, we have $d(0)=\sqrt {0^4-15(0^2)+64}=\sqrt{64}=8$
(c) For $x=1$, we have $d(1)=\sqrt {1^4-15(1)^2+64}=\sqrt{50}=5\sqrt {2}$
(d) Use a graphing tool to graph $d(x)=\sqrt{x^4-15x^2+64}$. Refer to the graph below.
(e) Using the graph generated by the graphing utility, locate the point/s closest to the origin and take their $x$-coordinate.. The points nearest the origin have the $x$-coordinates $\pm2.74$.
