Q. 45 - Precalculus Enhanced with Graphing Utilities

Question

List the intercepts and test for symmetry.

$y^2=x+4$

Step-by-Step Solution

Answer

• $(-4,0)$ is the x-intercept point.
• $(0,\pm 2)$ 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 -   $y^2=x+4$

Step 2. To Find

Find the intercept point and test for symmetry.

Step 3. Finding the x-intercept

Finding the x-intercept

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

$y^2=x+4$

Put $y=0$

$0=x+4$

$x=-4$

(-4,0) is the x-intercept.

Step 4. Finding the y-intercept

Finding the y-intercept

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

$y^2=x+4$

put $x=0$

$y^2=0+4$

$y=\pm 2$

$(0,\pm 2)$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 -

$y^2=x+4$

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

${(-y)}^2=x+4$

$y^2=x+4$

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 -

$y^2=x+4$

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

$y^2=(-x)+4$

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 -

$y^2=x+4$

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

${(-y)}^2=(-x)+4$

$y^2=(-x)+4$

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 $y^2=x+4$ will be -