Q. 3 - Precalculus Enhanced with Graphing Utilities

Question

3. Let $P=(x, y)$ be a point on the graph of $y=\sqrt{x}$.

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

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

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

Step-by-Step Solution

Answer

The distance $d$ is $d=\sqrt{x^2-x+1}$,the graph is 

 and $d$ is minimum when $x=0.5$

Part (a) Step 1: Given information

Given the point $P(x,y)$ and $(1,0)$

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

Substituting, we get

$d=\sqrt{(x-1{)}^2+y^2}$

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

Part (b) Step 1: Given information

Given the equation $d\left(x\right)=\sqrt{x^2-x+1}$

Part (b) Step 2: Sketching the graph

The graph is 

Part (c) Step 1: Given information

Given the equation $d\left(x\right)=\sqrt{x^2-x+1}$$d=\sqrt{x^2-x+1}$

Part (c) Step 2: Checking the graph

Checking, we get that $d$ is smallest when $x=0.5$