Question

2. Let $P=(x,y)$be a point on the graph of $y=x^2-8$.

(a) Express the distance $d$from $P$to the point $(0,-1)$as a function of $x$.

(b) What is $d$if $x=0$?

(c) What is $d$if $x=-1$?

(d) Use a graphing utility to graph $d=d\left(x\right)$.

(e) For what values of $x$is $d$smallest?

Short answer

The distance is $d=\sqrt{x^4-13x^2+49}$and the values are$7,6.08$and the graph is

and$d$is least when$$\begin{gathered} x=-2.55, \\ x=2.55 \end{gathered}$$

Step-by-step solution

Part (a) Step 1: Given information

Given the point$P(x,y)$and the curve$y=x^2-8$

Part (a) Step 2: Calculate the distance using the distance formula

The distance of $P$from $(0,-1)$is $d=\sqrt{x^2+{(y-1)}^2}$. So the required distance is

$$\begin{gathered} d\left(x\right)=\sqrt{x^2+{\left(x^2-8+1\right)}^2} \\ d\left(x\right)=\sqrt{x^2+{\left(x^2-7\right)}^2} \\ d\left(x\right)=\sqrt{x^2+\left(x^4-14x^2+49\right)} \\ d\left(x\right)=\sqrt{x^4-13x^2+49} \end{gathered}$$

Part (b) Step 1: Given information

Given the equation$d\left(x\right)=\sqrt{x^4-13x^2+49}$

Part (b) Step 2: Substituting x=0 and calculating the value

Substituting, we get

$$\begin{gathered} d\left(x\right)=\sqrt{x^4-13x^2+49} \\ d\left(0\right)=\sqrt{0^4-13{\left(0\right)}^2+49} \\ d\left(0\right)=\sqrt{49} \\ d\left(0\right)=7 \end{gathered}$$

Part (c) Step 1: Given information

Given the equation$d\left(x\right)=\sqrt{x^4-13x^2+49}$

Part (c) Step 2: Substituting x=1 and calculating the value

Substituting, we get

$$\begin{gathered} d\left(x\right)=\sqrt{x^4-13x^2+49} \\ d\left(1\right)=\sqrt{1^4-13{\left(1\right)}^2+49} \\ d\left(1\right)=\sqrt{37} \end{gathered}$$

Part (d) Step 1: Given information

Given the equation$d\left(x\right)=\sqrt{x^4-13x^2+49}$

Part (d) Step 2: Sketching the graph

The graph is

Part (e) Step 1: Given information

Given the equation$d\left(x\right)=\sqrt{x^4-13x^2+49}$

Part (e): Checking the graph

The value of$d$is least when$$\begin{gathered} x=-2.55, \\ x=2.55 \end{gathered}$$