Chapter 11.1, Problem 7E

To determine

To match: the equation with the graph.

Answer to Problem 7E

  

Explanation of Solution

Given information:

The equation is $x^2=-6y$ .

First write the equation in a standard form of parabola,

  $x^2-\left(-6y\right)$

Comparing the standard form of the parabola to the general equation,

  $x^2-4py$

By comparing the equations it can be observed that,

  $\begin{array}{l}4p=-6\\ p=-\frac{6}{4}\text{ [divide]}\\ p=-\frac{3}{2}\end{array}$

Directrix: the directrix of parabola is given by $y=-p$ ,

Since $p=-\frac{3}{2}$ , substitute it,

  $\begin{array}{l}y=-p\\ y=-\left(-\frac{3}{2}\right)\text{ [substitute p by -}\frac{3}{2}]\\ y=\frac{3}{2}\text{ [simplify]}\end{array}$

So the directrix of the parabola is $y=\frac{3}{2}$ .

Focus: the focus of parabola is given as $F\left(0,p\right)$ ,

Since $p=-\frac{3}{2}$ , substitute the value of $p$ ,

Thus, the focus of parabola is $F\left(0,-\frac{3}{2}\right)$ .

Vertices : $\left(0,0\right)$

Since the $\text{p<0}$ , so the parabola is opened downwards,

Use the above information together with some additional values which is show in table below

To sketch the graph,

y	x
-1	2.44
-2	3.46
-3	4.24
-4	4.89

The graph is obtained as: