Q. 50 - Precalculus Enhanced with Graphing Utilities

Question

List the intercepts and test for symmetry - 

$y=x^4-2x^2-8$

Step-by-Step Solution

Answer

• $(\pm 2,0)$ are the x-intercept point.
• $(0,-8)$ is the y- intercept point.
• The graph is not symmetric about x-axis.
• The graph is symmetric about y-axis.
• The graph is not symmetric about The origin.

Step 1. Given Data

Equation -   $y=x^4-2x^2-8$

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.

$y=x^4-2x^2-8$

Put  $y=0$

$0=x^4-2x^2-8$

$(x^2-4)(x^2+2)=0$

$(x-2)(x+2)(x^2+2)=0$

Since  $(x^2+2)$ can not be zero (0), then  $x=2 and (-2)$ are the only solution of the equation.

$(\pm 2,0)$ is the x-intercept. 

Step 4. Finding the y-intercept

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

$y=x^4-2x^2-8$

Put  $x=0$

$y=0^4-2\times {\left(0\right)}^2-8$

$y=-8$

(0,-8) 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 -

$y=x^4-2x^2-8$

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

$(-y)=x^4-2x^2-8$

$y=-x^4+2x^2+8$

This is not the same as the original equation, the graph will not 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 -

$y=x^4-2x^2-8$

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

$y={(-x)}^4-2{(-x)}^2-8$

$y=x^4-2x^2-8$

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 -

$y=x^4-2x^2-8$

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

$(-y)={(-x)}^4-2{(-x)}^2-8$

$-y=x^4-2x^2-8$

$y=-x^4+2x^2+8$

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 $y=x^4-2x^2-8$ will be -