Q. 1 - Precalculus Enhanced with Graphing Utilities

Question

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

(a) Express the distance $d$ from $P$ to the origin 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?

Step-by-Step Solution

Answer

The distance $d$ is $d\left(x\right)=\sqrt{x^4-15x^2+64}$ and the values are $8,7.07$ and the graph is 

and $d$ is smallest when $x=-2.74$ or

$x=2.74$

Part (a) Step 1: Given information

Given the point P  on the graph of $y=x^2-8$

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

The distance of $P$ from origin is $d=\sqrt{x^2+y^2}$. So the required distance is

$$\begin{gathered} d\left(x\right)=\sqrt{x^2+{\left(x^2-8\right)}^2} \\ d\left(x\right)=\sqrt{x^2+\left(x^4-16x^2+64\right)} \\ d\left(x\right)=\sqrt{x^4-15x^2+64} \end{gathered}$$

Part (b) Step 1: Given information

Given the equation $d\left(x\right)=\sqrt{x^4-15x^2+64}$

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

Substituting $x=0$, we get

$$\begin{gathered} d\left(x\right)=\sqrt{x^4-15x^2+64} \\ d\left(0\right)=\sqrt{0^4-15{\left(0\right)}^2+64} \\ d\left(0\right)=\sqrt{64} \\ d\left(0\right)=8 \end{gathered}$$

Part (c) Step 1: Given information

Given the equation $d\left(x\right)=\sqrt{x^4-15x^2+64}$

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

Substituting, we get

$$\begin{gathered} d\left(x\right)=\sqrt{x^4-15x^2+64} \\ d\left(1\right)=\sqrt{1^4-15{\left(1\right)}^2+64} \\ d\left(1\right)=\sqrt{1-15+64} \\ d\left(1\right)=\sqrt{50} \\ d\left(1\right)=7.07 \end{gathered}$$

Part (d) Step 1: Given information

Given the equation $d\left(x\right)=\sqrt{x^4-15x^2+64}$

Part (d) Step 2: Sketching the graph

Sketching, we get

Part (e) Step 1: Given information

Given the equation $d\left(x\right)=\sqrt{x^4-15x^2+64}$

Part (e) Step 2: Checking from graph

Checking, we get that $d$ is smallest when $x=-2.74$ or

$x=2.74$