Question

Graph: $\frac{x^2}{4}+\frac{y^2}{16}=1$.

Short answer

The graph of an ellipse is shown below:

Step-by-step solution

Given Information

The given ellipse is $\frac{x^2}{4}+\frac{y^2}{16}=1$.

Step 2. Write the equation in standard form.

The standard form of an ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\dots \left(1\right)$.

Compare the given equation with standard form and conclude that width="92">x24+y216=1is in standard form.

Step 3. Find whether the major axis is horizontal or vertical.

Compare the equation $\left(1\right)$with $\frac{x^2}{4}+\frac{y^2}{16}=1$to find $a^2$and $b^2$.

$\begin{array}{rcl}{a}^2& =& 4\\ b^2& =& 16\end{array}$

Since $16>4$and $16$is in the $y^2$term, the major axis will be vertical.

Step 4. Find the endpoints of the major axis.

The endpoints will be the $y$-intercepts.

Since $b^2=16$, $b=\pm 4$.

Thus, the coordinates of the endpoints of the major axis are role="math" localid="1645594892127" $(0,4)$and role="math" localid="1645594900062" $(0,-4)$.

Step 5. Find the endpoints of the minor axis.

The endpoints will be the $x$-intercepts.

Since $a^2=4$, $a=\pm 2$.

Thus, the coordinates of the endpoints of the minor axis are $(2,0)$and $(-2,0)$.

Step 6. Draw the ellipse.

The graph of an ellipse is shown below: