91Ó°ÊÓ

We want to find the vertices and foci of the ellipse given with its equation, then graph by hand and check that graph is correct using a graphing utility.

The equation we have to analyze in order to find the vertex and foci is

$4x^2+9y^2=36$

which written in standard form, by dividing with 36, is

$\frac{x^2}{9}+\frac{y^2}{4}=1$

First we, find the greater denominator. We can see that the grater denominator, 9, is located under the y-squared, therefore the ellipse has its major axis parallel to the y-axis.

Since the term squared have no numbers squared, only the variables x or y. the center of the ellipse is located in the origin (0,0).

Any ellipse equation has the following form:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1or\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$

but in both cases$a>b$

Since the greater denominator is 9 therefore it has to be that

localid="1647161324972" $$\begin{gathered} a^2=9⇒a=3 \\ and \\ {b}^2=4⇒b=2 \end{gathered}$$

An ellipse with its major axis parallel to the y-axis has vertices in the points

localid="1647161358898" $(0,-a)and(0,a)$

And since $a=3$the vertices of the given ellipse are

$(0,-3)and(0,3)$

The distance from the origin of the ellipse to the foci is c.

Looking at the following picture:

The triangle ABC is right triangle therefore the Pythagoras theorem holds and is used to find c when we have a and b.

$$\begin{gathered} c^2=a^2-b^2 \\ {c}^2=3^2-2^2 \\ {c}^2=9-4 \\ {c}^2=5 \\ c=\sqrt{5}≈2.24 \end{gathered}$$

The foci are also located on the major axis and have the following coordinates:

$(0,-c)and(0,c)$

Therefore they are$(0,-\sqrt{5})and(0,\sqrt{5})$

To graph the equation by hand first graph the found points: vertices, center of the ellipse and its x-intercepts (given by b)

The x-intercepts are$(-b,0)and(b,0)$

or when we plug in $b=2$we have$(-2,0)and(2,0)$

Now draw a cure representing the ellipse through the vertices and the y-intercepts and add the foci:

Since the graph doesn't pass the vertical line test, the equation of the ellipse is not a function.

In order to graph the ellipse we will need to graph two functions obtaining by first expressing $y^2$on the side and then taking square root to get two equations:

localid="1647162784710" $$\begin{gathered} \frac{y^2}{9}+\frac{x^2}{4}=1 \\ \frac{y^2}{9}=1-\frac{x^2}{4} \\ {y}^2=9-\frac{9x^2}{4} \end{gathered}$$

Finally we get the two equations we have to graph:

localid="1647162790991" $y=-\sqrt{9-\frac{9x^2}{4}}andy=\sqrt{9-\frac{9x^2}{4}}$