Chapter 1, Problem 13RE

To determine

To list: the intercepts and to test for symmetry

Answer to Problem 13RE

There is no $x$ -intercept and the y - intercept is (0,1) and the graph is symmetric with respect to the $y$ -axis.

Explanation of Solution

Given:

  $y=x^4+2x^2+1$

Calculation:

In order to find the $x$ -intercept, let $y=0$ and solve for $x$

  $x^4+2x^2+1=0$

Factor the equation.

  $x^4+2x^2+1={\left(x^2+1\right)}^2=0$

Use the square root property.

  $x^2+1=0$

  $x^2=-1$

Since the square of a number cannot be negative, there is no real solution for $x$ .

Therefore, there is no $x$ -intercept.

For finding the $y$ -intercept, let $x=0$ and solve for $y$ .

  $y=0+2\cdot 0+1$

  $y=1$

The $y$ -intercept is (0, 1)

Test for symmetry with respect to the $x$ -axis, $y$ -axis and the origin.

  $x-$ Axis:

For testing symmetry with respect to the $x$ -axis, replace $y$ by $-y$

  $-y=x^4+2x^2+1$

Since this equation is not equivalent to the original equation, the graph of the equation $y=x^4+$

  $2x^2+1$ is not symmetric with respect to the $x$ -axis.

  $y-Axis$ : For testing symmetry with respect to the $y$ -axis, replace $x$ by $-x$

  $y={(-x)}^4+2{(-x)}^2+1$

  $y=x^4+2x^2+1$

Since this equation is equivalent to the original equation, the graph of the equation $y=x^4+2x^2$

+1 is symmetric with respect to the $y$ axis.

Origin: For testing symmetry with respect to the origin, replace $x$ by $-x$ and

  $y$ By $-y$

  $-y={(-x)}^4+2{(-x)}^2+1$

  $-y=x^4+2x^2+1$

Multiply by -1 on both sides of the equation.

  $y=-x^4-2x^2-1$

Since this equation is not equivalent to the original equation, the graph of the equation $y=x^4+$

  $2x^2+1$ is not symmetric with respect to the origin.

Therefore the intercept is (0,1) and the graph is symmetric with respect to the $y$ -axis.

Conclusion:

There is no $x$ -intercept and the y - intercept is (0,1) and the graph is symmetric with respect to the $y$ -axis.