Q. 72 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 72, (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.

$x+y^2=4$

Step-by-Step Solution

Answer

The $x$-intercepts is $4$, and  and the $y$-intercept are $-2$ and $2$. The equation is symmetric with respect to the$x$ -axis.

The graph is shown below:

Step 1: Given information

The given equation is $x+y^2=4$.

Step 2: Determine the x -intercept.

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

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

Step 3: Determine the y -intercept.

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

$\begin{array}{rcl}0+y^2& =& 4\\ y^2& =& 4\\ y& =& 2,-2\end{array}$

Step 4: Determine the symmetry.

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

$\begin{array}{rcl}x+{\left(-y\right)}^2& =& 4\\ x+y^2& =& 4\end{array}$

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

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

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

As the obtained function is not equivalent to the given function, there is not 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.

$\begin{array}{rcl}-x+{\left(-y\right)}^2& =& 4\\ -x+y^2& =& 4\\ x-y^2& =& -4\end{array}$

As the obtained function is not equivalent to the given function, there is no symmetry about the origin.

Step 5: Graph the equation

Substitute different values of $x$ into the given function to find the corresponding values of $y$. Represent the data in tabular form.

$y$	$x$	$\left(x,y\right)$
$-3$	$4-{\left(-3\right)}^2=-5$	$\left(-5,-3\right)$
$-2$	$4-{\left(-2\right)}^2=0$	$\left(0,-2\right)$
$-1$	$4-{\left(-1\right)}^2=3$	$\left(3,-1\right)$
$1$	$4-{\left(1\right)}^2=3$	$\left(3,1\right)$
$2$	$4-{\left(2\right)}^2=0$	$\left(0,2\right)$
$3$	$4-{\left(3\right)}^2=-5$	$\left(-5,3\right)$

The graph is shown below:

Step 6: Verify the graph using graphing calculator.

Solve for $y$.

$\begin{array}{rcl}{y}^2& =& 4-x\\ y& =& \pm \sqrt{4-x}\end{array}$

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

Step 7: Write the conclusion

The $x$-intercept is $\left(4,0\right)$.

The $y$-intercept are $\left(0,-2\right)$ and $\left(0,2\right)$ .

The graph is symmetric with respect to $x$-axis.

The graph is shown below: