Chapter 1.2, Problem 60AYU

In Problems $57-72$, list the intercepts and test for symmetry.

$x^2-y-4=0$

To determine

The intercepts and symmetry for the equation $x^2-y-4=0$.

Answer to Problem 60AYU

Solution:

The $x$-intercepts are $(2,0)$ and $(-2,0)$, and $y$-interceptis $(0,-4)$.

The graph of the equation $x^2-y-4=0$ is symmetric with respect to $y$ -axis, but it is notsymmetric with respect to $x$ -axis and origin.

Explanation of Solution

Given Information:

The equation $x^2-y-4=0$.

Explanation:

The points, if any, at which a graph crosses or touches the coordinate axes, are called the intercepts of the graph.

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

$\Rightarrow x^2-0-4=0$

$\Rightarrow x^2=4$

$\Rightarrow x=\pm 2$.

Therefore, $x$-intercepts are $(2,0)$ and $(-2,0)$.

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

$\Rightarrow 0^2-y-4=0$

$\Rightarrow y=-4$.

The $y$-interceptis $(0,-4)$.

To test for symmetry with respect to $x$-axis, replace $y$ by $-y$ and if itresults inan equation equivalent to the original equation, then the equation is symmetric with respect to $x$-axis.

$\Rightarrow$ $x^2-(-y)-4=0$ is not equivalent to the $x^2-y-4=0$.

Hence, the graph of the equation $x^2-y-4=0$ is not symmetric with respect to $x$-axis.

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

$\Rightarrow {(-x)}^2-y-4=0$ is equivalent to $x^2-y-4=0$.

Hence, the graph of the equation $x^2-y-4=0$ is symmetric with respect to $y$-axis.

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

$\Rightarrow {(-x)}^2-(-y)-4=0$

$\Rightarrow x^2+y-4=0$ is not equivalent to $x^2-y-4=0$.

Hence, the graph of the equation $x^2-y-4=0$ is not symmetric with respect to origin.