Q. 73 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 73, (a) find the intercepts of each equation, (b) test each equation for symmetry with respect to the x-axis, the y-axis, and the origin, and (c) graph each equation by hand by plotting points. Be sure to label the intercepts on the graph and use any symmetry to

assist in drawing the graph. Verify your results using a graphing utility.

$x^2+y^2=9$

Step-by-Step Solution

Answer

The $x$-intercepts are -3, and 3 and the $y$-intercept are -3 and 3. The equation is symmetric with respect to the $y$-axis, $x$-axis and origin.

The graph is shown below:

Step 1: Given information

The given equation is $x^2+y^2=9$

Step 2: Determine the x and y intercept.

For $y$-intercept, substitute $0$ for $x$ in the given equation.

$\begin{array}{rcl}{0}^2+y^2& =& 9\\ y& =& \pm 3\end{array}$

For $x$-intercept, substitute $0$ for $y$ in the given equation.

$\begin{array}{rcl}{x}^2+0^2& =& 9\\ x& =& \pm 3\end{array}$

Plot the $x$ and $y$-intercept in the graph.

Step 3: Determine the symmetry

Replace $y$ with $-y$ in the given function and simplify to check the symmetry about the $x$-axis.

$\begin{array}{rcl}{x}^2+{\left(-y\right)}^2& =& 9\\ x^2+y^2& =& 9\end{array}$

As the obtained function is equivalent to the given function, there is symmetry about the $x$-axis.

Replace $x$ with $-x$ in the given function and simplify to check the symmetry about the

$\begin{array}{rcl}{\left(-x\right)}^2+y^2& =& 9\\ x^2+y^2& =& 9\end{array}$

As the obtained function is equivalent to the given function, there is a symmetry about the $y$-axis.

Replace $y$ with $-y$ and $x$ with $-x$ in the given function and simplify to check the symmetry about the origin.

$\begin{array}{rcl}{\left(-x\right)}^2+{\left(-y\right)}^2& =& 9\\ x^2+y^2& =& 9\end{array}$

As the obtained function is equivalent to the given function, there symmetry about the origin.

Step 4: Verify the graph using graphing utility.

By using the graphing utility, the graph is shown below: 

Step 5: Write the conclusion

The $x$-intercept are (-3,0) and (3,0).

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

The graph is symmetric with respect to $x$-axis, $y$-axis and origin.

The graph is shown below: