Chapter 7, Problem 3CR

To determine

The symmetry and the intercepts for the equation $3x+y^2=9$.

Answer to Problem 3CR

Solution:

The graph of the equation $3x+y^2=9$ is symmetric with respect to $x$ -axis, but not

with respect to $y$ -axis and origin.

The $x$-intercept is $(3,0)$ and the $y$-intercepts are $(0,3)$ and $(0,-3)$.

Explanation of Solution

Given information:

The equation $3x+y^2=9$.

Explanation:

To test for symmetry with respect to $x$-axis, replace $y$ by $-y$. If it results equivalent to the original equation, then the equation is symmetric with respect to $x$-axis.

$\Rightarrow$ $3x+{(-y)}^2=9$ is equivalent to the $3x+y^2=9$.

Hence, the graph of the equation $3x+y^2=9$ is symmetric with respect to $x$-axis.

To test for symmetry with respect to $y$-axis, replace $x$ by – $x$. If it results equivalent to the original equation, then the equation is symmetric with respect to $y$-axis.

$\Rightarrow 3\left(-x\right)+y^2=9$ is not equivalent to $3x+y^2=9$.

Hence, the graph of the equation $3x+y^2=9$ is not symmetric with respect to $y$-axis.

To test for symmetry with respect to the origin, replace $x$ by – $x$ and $y$ by $-y$. If the results are equivalent to the original equation, then the equation is symmetric with respect to the origin.

$\Rightarrow 3\left(-x\right)+{\left(-y\right)}^2=9$.

$\Rightarrow -3x+y^2=9$ is not equivalent to $3x+y^2=9$.

Hence, the graph of the equation $3x+y^2=9$ is not symmetric with respect to the origin.

The intercepts of an equation are obtained by using the information that points on the $x$-axis have

$y$-coordinates equal to $0$, and the point on the $y$-axis have $x$-coordinates equal to $0$.

To find $x$-intercept(s), set $y=0$.

$\Rightarrow 3x=9$

$\Rightarrow x=3$.

Therefore, the $x$-intercept is $(3,0)$.

To find the $y$-intercept(s), let $x=0$.

$\Rightarrow 3\left(0\right)+y^2=9$

$\Rightarrow y^2=9$

$\Rightarrow y=\pm 3$.

The $y$-intercepts are $(0,3)$ and $(0,-3)$.