Chapter 3.6, Problem 1AE

To determine

To answer: The questions related to the function, $y=x^2-8$.

Answer to Problem 1AE

a. $d\left(x\right)=\sqrt{x^4-15x^2+64}$

Explanation of Solution

Given:

$y=x^2-8$

$P=\left(x, y\right)$

Calculation:

a. Distance $d$ as a function of $x$

Let, $P=\left(x, y\right)$ be the point on the graph of $y=x^2-8$. The distance from $O$ to $P$ is

$d=\sqrt{{\left(x-0\right)}^2+{\left(y-0\right)}^2}$

$=\sqrt{{\left(x\right)}^2+{\left(y\right)}^2}$

Substituting $y=x^2-8$ for $y$, then

$d\left(x\right)=\sqrt{{\left(x\right)}^2+{\left(x^2-8\right)}^2}$

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

To determine

To answer: The questions related to the function, $y=x^2-8$.

Answer to Problem 1AE

b. $d\left(0\right)=8$

Explanation of Solution

Given:

$y=x^2-8$

$P=\left(x, y\right)$

Calculation:

b. Value of $d$ if $x=0$

Substituting $x=0$, the distance $d$ is

$d\left(0\right)=\sqrt{0^4-15{\left(0\right)}^2+64}$

$d\left(0\right)=8$

To determine

To answer: The questions related to the function, $y=x^2-8$.

Answer to Problem 1AE

c. $d\left(1\right)=7.07$

Explanation of Solution

Given:

$y=x^2-8$

$P=\left(x, y\right)$

Calculation:

c. Value of $d$ if $x=1$

Substituting $x=1$, the distance $d$ is

$d\left(1\right)=\sqrt{1-15+64}$

$d\left(1\right)=\sqrt{50}\cong 7.07$

To determine

To answer: The questions related to the function, $y=x^2-8$.

Answer to Problem 1AE

Explanation of Solution

Given:

$y=x^2-8$

$P=\left(x, y\right)$

Calculation:

d. Graph

To determine

To answer: The questions related to the function, $y=x^2-8$.

Answer to Problem 1AE

(e) $x=-2.74$ or $x=2.74$

Explanation of Solution

Given:

$y=x^2-8$

$P=\left(x, y\right)$

Calculation:

(e) Smallest $d$

The value of $d$ is smallest when $x=-2.74$ or $x=2.74$.