Chapter 1.2, Problem 49AYU

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

$y^2=x+4$

To determine

To find: The intercepts and test for the symmetry.

Answer to Problem 49AYU

$x\text{-intercept}$ is $-4$ and $y\text{-intercepts}$ are $-2$ and 2. The graph of the equation is symmetric with respect to $x\text{-axis}$.

Explanation of Solution

Given:

The equation $y^2=x+4$

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

Therefore   $\text{<mi>x</mi> + 4 = 0}$.

$x\text{ + 4 <mo>−</mo> 4 = 0 <mo>−</mo> 4}$   Subtract 4 from both sides

$\text{<mi>x</mi> = <mo>−</mo>4}$

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

Therefore   $y^2\text{ = }\left(\text{0}\right)\text{ + 4}$

$y^2\text{ = 4}$

$y =\pm 2$

The $x\text{-intercept}$ is $-4$ and $y\text{-intercepts}$ are $-2$ and 2.

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

${(-y)}^2\text{ = <mi>x</mi> + 4}$

$y^2 = x + 4$. 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 $-x$

$y^2=(-x)+4$ is not equivalent to $y^2=x+4$

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 $-y$.

${(-y)}^2\text{ = (<mo>−</mo><mi>x</mi>) + 4}$. $y^2\text{ = <mo>−</mo><mi>x</mi> + 4}$ which is not equal to the original equation. Therefore the graph of the equation is not symmetric with respect to the origin.