Q. 74 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 74, (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.

$x^2+y^2=16$

Step-by-Step Solution

Answer

1. The $x$-intercepts are $-4$ and 4 and the y-intercepts are -4 and 4 .
2. The equation is symmetric with respect to the x-axis, y-axis and origin.
3. The graph is shown below:

Step 1: Given information

The giving equation is $x^2+y^2=16$

Step 2: Determine the x and y intercept.

For $y$-intercept, substitute $0$ for $x$ and then solve for $y$.

$\begin{array}{rcl}{x}^2+y^2& =& 16\\ 0^2+y^2& =& 16\\ y& =& \pm 4\end{array}$

For $x$-intercept, substitute $0$ for $y$ and then solve for $x$.

$\begin{array}{rcl}{x}^2+y^2& =& 16\\ y^2+0& =& 16\\ y& =& \pm 4\end{array}$

The graph is shown below:

Step 3: Determine the symmetry

Perform the test for symmetry with respect to the $x$-axis. Put $y=-y$

$\begin{array}{rcl}{x}^2+{\left(-y\right)}^2& =& 16\\ x^2+y^2& =& 16\end{array}$

Since after substituting $y=-y$ into the given equation, the resulting equation is equivalent to the original equation hence the graph of the given equation is symmetrical about the $x$-axis.

Perform the test for symmetry with respect to the $y$-axis. Put $x=-x$

$\begin{array}{rcl}{\left(-x\right)}^2+y^2& =& 16\\ x^2+y^2& =& 16\end{array}$

Since after substituting $x=-x$ into the given equation, the resulting equation is equivalent to the original equation hence the graph of the given equation is symmetrical about the  $y$-axis.

Perform the test for symmetry with respect to the origin. Put $x=-x ; y=-y$

$\begin{array}{rcl}{\left(-x\right)}^2+{\left(-y\right)}^2& =& 16\\ x^2+y^2& =& 16\end{array}$

Since after substituting $x=-x;y=-y$ into the given equation, the resulting equation is equivalent to the original equation hence the graph of the given equation is symmetrical about the origin.

Step 4: Verify the graph using graphing calculator.

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

Step 5: Write the conclusion

1. The x-intercept and y-intercept are $\left(0,4\right),\left(0,-4\right),\left(4,0\right)$ and $\left(-4,0\right)$.
2. The graph is symmetric with respect to x-axis, y-axis and origin.
3. The graph is shown below: