Q. 52 - Precalculus Enhanced with Graphing Utilities

Question

List the intercepts and test for symmetry -

$4x^2+y^2=4$

Step-by-Step Solution

Answer

• $(\pm 1,0)$ are the x-intercept point.
• $(0,\pm 2)$ are the x-intercept point.
• The graph is symmetric about x-axis.
• The graph is symmetric about y-axis.
• The graph is symmetric about the origin.

Step 1. Given Data

Equation -   $4x^2+y^2=4$

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.

$4x^2+y^2=4$

Put  $y=0$

$4x^2+0^2=4$

$4x^2=4$

$x^2=1$

$x=\pm 1$

$(\pm 1,0)$ is the x-intercept. 

Step 4. Finding the y-intercept

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

$4x^2+y^2=4$

Put  $x=0$

$4\times 0^2+y^2=4$

$y^2=4$

$y=\pm 2$

$(0,\pm 2)$ is 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 -

$4x^2+y^2=4$

Replacing (y) with (-y) gives us - 

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

$4x^2+y^2=4$

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 -

$4x^2+y^2=4$

Replacing (x) with (-x) gives us - 

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

$4x^2+y^2=4$

This is same as the original equation, the graph will 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 -

$4x^2+y^2=4$

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

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

$4x^2+y^2=4$

This is same as the original equation, the graph will be symmetric about the the origin.

Step 6. Graph

The graph of equation $4x^2+y^2=4$ will be -