Q. 46 - Precalculus Enhanced with Graphing Utilities

Question

List the intercepts and test for symmetry -

$y^2=x+9$

Step-by-Step Solution

Answer

• $(-9,0)$ is the x-intercept point.
• $(0,\pm 3)$ is the y-intercept point.
• The graph is symmetric about x-axis.
• The graph is not symmetric about y-axis.
• The graph is not symmetric about the origin. 

Step 1. Given Data

Equation - $y^2=x+9$

Step 2. To Find

Find the intercept point and test for symmetry.

Step 3. Finding the x-intercept

To find the x-intercepts, replace y with 0 and solve for x.

$y^2=x+9$

Put $y=0$

$0=x+9$

$x=-9$

(-9,0) is the x-intercept.

Step 4. Finding the y-intercept

To find the x-intercepts, replace x with 0 and solve for y.

$y^2=x+9$

Put $x=0$

$y^2=0+9$

$y^2=9$

$y=\pm 3$

$(0,\pm 3)$ are the y-intercept 

Step 5. Check for symmetry

• Checking symmetry about x-axis

if replacing (y) with (-y) leaves the equation unchanged, the graph will be symmetric about the x-axis.

we will start with original equation -

$y^2=x+9$

Replacing (y) with (-y) gives us - 

${(-y)}^2=x+9$

$y^2=x+9$

This is same as the original equation, the graph will be symmetric about the x-Axis.

• Checking symmetry about y-axis

if replacing (x) with (-x) leaves the equation unchanged, the graph will be symmetric about the y-axis.

we will start with original equation -

$y^2=x+9$

Replacing (x) with (-x) gives us - 

$y^2=(-x)+9$

This is not the same as the original equation, the graph will not be symmetric about the y-Axis.

• Checking symmetry about the origin

if replacing (x) with (-x) and (y) with (-y)leaves the equation unchanged, the graph will be symmetric about the origin.

we will start with original equation -

$y^2=x+9$

Replacing (x) with (-x) and (y) with (-y) gives us - 

${(-y)}^2=(-x)+9$

$y^2=(-x)+9$

This is not the same as the original equation, the graph will not be symmetric about the origin.

Step 6. Graph

The graph of equation $y^2=x+9$ will be -