Chapter 1, Problem 11RE

To determine

To list: the intercepts and to test for symmetry

Answer to Problem 11RE

Therefore, the intercepts are (4,0),(-4,0),(0,-2) and (0,2) and the graph is symmetric with respect to the $x$ -axis, $y$ -axis and the origin.

Explanation of Solution

Given:

  $x^2+4y^2=16$

Calculation:

In order to find the $x$ -intercept, let $y=0$ and solve for $x$

  $x^2=16$

Subtract 16 from both the sides.

  $x^2-16=16-16$

  $x^2-16=0$

Factor the equation $x^2-16=0$

  $x^2-4^2=(x+4)(x-4)=0$

Use the zero product property.

  $\begin{array}{l}x+4=0\\ x=-4\\ x-4=0\\ x=4\end{array}$

The $x$ -intercepts are (4,0) and (-4,0) .

For finding the $y$ -intercept, let $x=0$ and solve for $y$ .

  $4y^2=16$

Subtract 16 from both the sides.

  $4y^2-16=16-16$

  $4y^2-16=0$

Factor out the common factor 4

  $4\left(y^2-4\right)=0$

Factor the equation $x^2-4=0$

  $x^2-2^2=(x+2)(x-2)=0$

Use the zero product property.

  $x+2=0$

  $x=-2$

  $x-2=0$

  $x=2$

The $y$ -intercepts are (0,-2) and (0,2) .

Test for symmetry with respect to the $x$ -axis, $y$ -axis and the origin.

  $x-Axis$ . For testing symmetry with respect to the $x$ -axis, replace $y$ by $-y$ .

  $\begin{array}{r}\hfill x^2+4{(-y)}^2=16\\ \hfill x^2+4y^2=16\end{array}$

Since this equation is equivalent to the original equation, the graph of the equation $x^2+4y^2=16$ is symmetric with respect to the $x$ -axis.

  $y-Axis$ . For testing symmetry with respect to the $y$ -axis, replace $x$ by $-x$ .

  $\begin{array}{r}\hfill {(-x)}^2+4y^2=16\\ \hfill x^2+4y^2=16\end{array}$

Since this equation is equivalent to the original equation, the graph of the equation $x^2+4y^2=16$ is symmetric with respect to the $y$ -axis.

Origin: For testing symmetry with respect to the origin, replace $x$ by $-x$ and

  $y$ by $-y$

  $\begin{array}{r}\hfill {(-x)}^2+4{(-y)}^2=16\\ \hfill x^2+4y^2=16\end{array}$

Since this equation is equivalent to the original equation, the graph of the equation $x^2+4y^2=16$ is symmetric with respect to the origin.

Therefore, the intercepts are (4,0),(-4,0),(0,-2) and (0,2) and the graph is symmetric with respect to the $x$ -axis, $y$ -axis and the origin.

Conclusion:

Therefore, the intercepts are (4,0),(-4,0),(0,-2) and (0,2) and the graph is symmetric with respect to the $x$ -axis, $y$ -axis and the origin.