Chapter 9.3, Problem 8OE

To determine

a)

To find: The vertex of parabola.

Answer to Problem 8OE

The vertex of the given parabola is $\left(0,0\right)$ .

Explanation of Solution

Given information: The equation of parabola $x=-\frac{1}{12}{y}^2$

Formula used: $x-h=a{\left(y-k\right)}^2$

Calculation: Rearrange the given equation,

  $\begin{array}{l}x=-\frac{1}{12}{y}^2\\ x-0=-\frac{1}{12}\times {\left(y-0\right)}^2\end{array}$

Compare $x-0=-\frac{1}{12}\times {\left(y-0\right)}^2$ to $x-h=a{\left(y-k\right)}^2$

Hence, the vertex $\left(h,k\right)=\left(0,0\right)$ .

Therefore, the required vertex is $\left(0,0\right)$ .

To determine

b)

To find: The value of $\left|c\right|$ .

Answer to Problem 8OE

The value of $\left|c\right|$ is $3$ .

Explanation of Solution

Given information: The equation of parabola $x=-\frac{1}{12}{y}^2$

Formula used: $x-h=a{\left(y-k\right)}^2$

Calculation: Rearrange the given equation,

  $\begin{array}{l}x=-\frac{1}{12}{y}^2\\ x-0=-\frac{1}{12}\times {\left(y-0\right)}^2\end{array}$

Compare $x-0=-\frac{1}{12}\times {\left(y-0\right)}^2$ to $x-h=a{\left(y-k\right)}^2$

Hence,

  $\begin{array}{l}\because a=-\frac{1}{12}\&\left|c\right|=\left|\frac{1}{4a}\right|\\ \therefore \left|c\right|=\left|\frac{1}{4\times \left(-\frac{1}{12}\right)}\right|\\ =\left|\frac{1}{-\frac{1}{3}}\right|\\ \therefore \left|c\right|=3\end{array}$

Therefore, the value of $\left|c\right|$ is $3$ .

To determine

c)

To find: The direction of opening of the parabola.

Answer to Problem 8OE

The parabola opens left.

Explanation of Solution

Given information: The equation of parabola $x=-{\left(y-2\right)}^2$

Formula used: $x-h=a{\left(y-k\right)}^2$

Calculation: Given information: The equation of parabola $x=-\frac{1}{12}{y}^2$

Formula used: $x-h=a{\left(y-k\right)}^2$

Calculation: Rearrange the given equation,

  $\begin{array}{l}x=-\frac{1}{12}{y}^2\\ x-0=-\frac{1}{12}\times {\left(y-0\right)}^2\end{array}$

Compare $x-0=-\frac{1}{12}\times {\left(y-0\right)}^2$ to $x-h=a{\left(y-k\right)}^2$

Hence,

  $\begin{array}{l}\therefore a=-\frac{1}{12}\\ \therefore a<0\end{array}$

Therefore, the direction of opening of the parabola will be left.