Chapter 1.2, Problem 54AYU

In Problems 49-64, list the intercepts and test for symmetry.

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

To determine

To find: The intercepts and test for the symmetry.

Answer to Problem 54AYU

$x\text{-intercepts}$ are $-2$ and 2 and $y\text{-intercept}$ is $-8$. The graph of the equation is symmetric with respect to $y\text{-axis}$.

Explanation of Solution

Given:

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

To find the $x\text{-intercept(s)}$, let $y\text{ = 0}$

Therefore   $x^4 - 2x^2 - 8 = 0$

$x^4 - 4x^2 + 2x^2 - 8 = 0$   Splitting the middle term

$x^2(x^2 - 4) + <mn>2</mn>(x^2 - 4) = 0$   Grouping the terms and taking common factors

$(x^2 - 4)(x^2 + 2) = 0$   Taking the common factor $(x^2 - 4)$

$x^2 - 4 = 0$   or   $x^2 + 2 = 0$

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

$x^2 = 4$   or   $x^2 = -2$

$x=\pm 2$   or   $x=\pm \sqrt{-2}$ not real

The $x\text{-intercepts}$ are 2 and $-2$

To find the $y\text{-intercept(s)}$, let $x\text{ = 0}$

Therefore   ${y\text{ = (0)}}^{\text{4}} {-\text{ 2(0)}}^{\text{2}}\text{ <mo>−</mo><mo> </mo>8}$

$y\text{ = <mo>−</mo>8}$

The $y\text{-intercept}$ is $-8$

Now test the equation for symmetry with respect to $x\text{-axis}$. Replace $y$ by $-y$.

$\text{(<mo>−</mo><mi>y</mi>)} = x^4 - 2x^2 - 8$

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

The equation is not same when $y$ is replaced by $-y$. Therefore the graph of the equation is not symmetric with respect to $x\text{-axis}$.

Now test the equation for symmetry with respect to $y\text{-axis}$. Replace $x$ by $-x$.

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

$y = x^4 - 2x^2 - 8$ is equivalent to the original equation.

Therefore the graph of the equation is symmetric with respect $y\text{-axis}$.

Now test the equation for symmetry with respect to the origin. Replace $x$ by $-x$ and $y$ by $-y$.

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

$-y = x^4 - 2x^2 - 8$ is not equal to the original equation. Therefore the graph of the equation is not symmetric with respect to the origin.