Chapter 1, Problem 9CT

To determine

The intercepts and test for symmetry for an equation $x^2+y=9$ .

Answer to Problem 9CT

Solution:

  $x-$ intercepts are $3,-3$, and $y-$ intercept is $9$ .

The graph of an equation $x^2+y=9$ is symmetric about the$y-$axis.

Explanation of Solution

Given information:

An equation $x^2+y=9$ .

To determine the $x$ -intercepts, substitute $y=0$ in the equation $x^2+y=9$, and solve for $x$ .

By substituting $y=0$ in $x^2+y=9$, it gives

  $x^2+0=9$ .

By simplifying,

  $\Rightarrow x^2=9$ .

By taking square root,

  $\Rightarrow x=\pm 3$ .

Therefore, the $x$ - intercepts are $3$ and $-3$ .

To determine the $y$ -intercepts, substitute $x=0$ in the equation $x^2+y=9$, and solve for $y$ .

By substituting $x=0$ in $x^2+y=9$, it gives

  ${\left(0\right)}^2+y=9$ .

By simplifying,

  $\Rightarrow 0+y=9$ .

  $\Rightarrow y=9$ .

Therefore, the $y$ - intercept is $9$ .

Now, test for symmetry:

For $x-$ axis: Replace $y$ by $-y$ in the equation $x^2+y=9$ .It gives $x^2-y=9$, which is not equal to $x^2+y=9$ .

For $y-$ axis: Replace $x$ by $-x$ in the equation $x^2+y=9$ .It gives ${\left(-x\right)}^2+y=9$, which is equivalent to $x^2+y=9$ . So, the graph of equation $x^2+y=9$ is symmetric with respect to $y-$ axis.

For origin: Replace $x$ by $-x$ and $y$ by $-y$ in the equation $x^2+y=9$ .It gives ${\left(-x\right)}^2+\left(-y\right)=9$, which is not equal to $x^2+y=9$ .

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