Chapter 1.2, Problem 50AYU

In Problems 49-64, list the intercepts and test for symmetry.

$y^2=x+9$

To determine

To find: The intercepts and test for the symmetry.

Answer to Problem 50AYU

$x\text{-intercept}$ is $-9$ and $y\text{-intercepts}$ are $-3$ and 3. The graph of the equation is symmetric with respect to $x\text{-axis}$.

Explanation of Solution

Given:

The equation $y^2=x+9$

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

Therefore   $x\text{ + 9 = 0}$.

$x\text{ + 9 <mo>−</mo> 9 = 0 <mo>−</mo> 9}$   Subtract 9 from both sides

$x\text{ = <mo>−</mo>9}$

To find the $y\text{-intercept(s)}$, let $x\text{ = 0}$

Therefore   $y^2\text{ = }\left(\text{0}\right)\text{ + 9}$

$y^2\text{ = 9}$

$y=\pm 3$

The $x\text{-intercept}$ is $-9$ and $y\text{-intercepts}$ are $-3$ and 3.

Now test the equation for symmetry with respect to $x\text{-axis}$. Replace $y$ by $-y$.

${(-y)}^2\text{ = <mi>x</mi> + 9}$

$y^2 = x + 9$. The equation is same when $y$ is replaced by $-y$. Therefore the graph of the equation is symmetric with respect to $x\text{-axis}$.

Now test the equation for symmetry with respect to $y\text{-axis}$. Replace $x$ by $-x$

$y^2\text{ = (<mo>−</mo><mi>x</mi>) + 9}$ is not equivalent to $y^2=x+9$

Therefore the graph of the equation is not symmetric with respect $y\text{-axis}$.

Now test the equation for symmetry with respect to the origin. Replace $x$ by $-x$ and $y$ by $-y$.

${(-y)}^2=\left(-x\right)+9$. $y^2\text{= <mo>−</mo><mi>x</mi> + 9}$ which is not equal to the original equation. Therefore the graph of the equation is not symmetric with respect to the origin.