Chapter 1.9, Problem 29E

To determine

To graph: The ellipse $4x^2+2y^2=1$ by solve for $y$ and graphing two equations corresponding to the negative and positive square roots.

Explanation of Solution

Given information:

The equation of ellipse $4x^2+2y^2=1$ .

Graph:

The graph of ellipse $4x^2+2y^2=1$ by solving for $y$ .

Subtract $4x^2$ from the equation of ellipse,

  $\begin{array}{l}4x^2-4x^2+2y^2=1-4x^2\\ 2y^2=1-4x^2\end{array}$

Divided by 2 in the equation,

  $y^2=\frac{1-4x^2}{2}$

Take under root on both side of the equation,

  $y=\pm \sqrt{\frac{1-4x^2}{2}}$

From the two equations graph of ellipse is described:

  $y=+\sqrt{\frac{1-4x^2}{2}},y=-\sqrt{\frac{1-4x^2}{2}}$

The graph of the equation $y=\sqrt{\frac{1-4x^2}{2}}$ represents upper half of the ellipse because y is always greater than or equal to 0.

  $\begin{array}{c}\sqrt{\frac{1-4x^2}{2}}\ge 0\\ 1-4x^2\ge 0\\ 1\ge 4x^2\\ \pm \frac{1}{2}\ge x\end{array}$

So, the values of $x$ is less than or equal to $\pm \frac{1}{2}$ .

Thus, the graph appropriate in the $\left[-\frac{1}{2}\text{,}\frac{1}{2}\right]\times \left[\text{0,1}\right]$ viewing rectangle

The graph of the equation $y=\sqrt{\frac{1-4x^2}{2}}$ can be sketched using the table,

$x$	$y$
0	$\sqrt{\frac{1}{2}}$
$\frac{1}{8}$	$\sqrt{\frac{15}{32}}$
$\frac{1}{4}$	$\sqrt{\frac{3}{8}}$

The graph of equation is provided below,

  

The graph of the equation $y=-\sqrt{\frac{1-4x^2}{2}}$ represents lower half of the ellipse because y is always less than or equal to 0.

  $\begin{array}{c}-\sqrt{\frac{1-4x^2}{2}}\le 0\\ \frac{1-4x^2}{2}\le 0\\ 1-4x^2\le 0\\ \pm \frac{1}{2}\le x\end{array}$

So, the values of $x$ is greater than or equal to $\pm \frac{1}{2}$ .

Thus, the graph appropriate in the $\left[-\frac{1}{2}\text{,}\frac{1}{2}\right]\times \left[\text{-1,0}\right]$ viewing rectangle

The graph of the equation $y=-\sqrt{\frac{1-4x^2}{2}}$ can be sketched using the table,

$x$	$y$
0	$-\sqrt{\frac{1}{2}}$
$\frac{1}{2}$	0
$\frac{1}{4}$	$-\sqrt{\frac{3}{8}}$

The graph of equation is provided below,

  

the graph of ellipse $4x^2+2y^2=1$ appropriate in the $\left[-\frac{1}{2}\text{,}\frac{1}{2}\right]\times \left[\text{-1,1}\right]$ viewing rectangle

The graph of both equations in same figure is provided below,

  

Interpretation:

The graph of an equation in a viewing screen is a viewing rectangle.

The $\text{x}$ -values to range from a minimum value of $\text{xmin}$ $=\text{a}$ to a maximum value of $\text{xmax}$ $=\text{b}$

The $\text{y}$ -values to range from minimum value of $\text{ymin}$ $=\text{c}$ to a maximum value of $\text{ymax}$ $=\text{d}$

Then, the display portion of the graph lies in the rectangle $\left[\text{a,b}\right]\times \left[\text{c,d}\right]=\left\{\left(\text{x,y}\right)|\text{a}\le \text{x}\le \text{b},\text{c}\le \text{y}\le \text{d}\right\}$

The graph of $y=\sqrt{\frac{1-4x^2}{2}}$ appropriate in the $\left[-\frac{1}{2}\text{,}\frac{1}{2}\right]\times \left[\text{0,1}\right]$ viewing rectangle and graph make upper half ellipse.

The graph of $y=-\sqrt{\frac{1-4x^2}{2}}$ appropriate in the $\left[-\frac{1}{2}\text{,}\frac{1}{2}\right]\times \left[\text{-1,0}\right]$ viewing rectangle and graph make lower half ellipse.