Q. 49 - Precalculus Enhanced with Graphing Utilities

Question

List the intercepts and test for symmetry - 

$y=x^4-8x^2-9$

Step-by-Step Solution

Answer

• $(\pm 3,0)$ are the x-intercept point.
• $(0,-9)$ 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 -  $y=x^4-8x^2-9$

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. 

$y=x^4-8x^2-9$

Put  $y=0$

$0=x^4-8x^2-9$

$(x^2-9)(x^2+1)=0$

$(x^2-3^2)(x^2+1)=0$

$(x^2-3)(x^2+3)(x^2+1)=0$

Since  $(x^2+1)$ can not be zero (0), then $x=3 and (-3)$ are the only solution of the equation.

(-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.

$y=x^4-8x-9$

Put  $x=0$

$y=0^4-8\times 0-9$

$y=-9$

(0,-9) 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 -

$y=x^4-8x-9$

Replacing (y) with (-y) gives us -  

$-y=x^4-8x-9$

$y=-x^4+8x+9$

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 -

$y=x^4-8x^2-9$

Replacing (x) with (-x) gives us -  

$y={(-x)}^4-8{(-x)}^2-9$

$y=x^4-8x^2-9$

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 -

$y=x^4-8x^2-9$

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

$(-y)={(-x)}^4-8{(-x)}^2-9$

$-y=x^4-8x^2-9$

$y=-x^4+8x^2+9$

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 $y=x^4-8x^2-9$ will be -