Q 12 - Precalculus Enhanced with Graphing Utilities

Question

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

$y=x^4-3x^2-4$.

Step-by-Step Solution

Answer

The given equation is symmetric about y-axis.

But it is not symmetric about x-axis and origin.

Step 1. Given Information

We have given the following equation :-  

$y=x^4-3x^2-4$.

We have to check the symmetry of this equation with respect to x-axis, y-axis and the origin.  

Step 2. To check symmetry about x-axis.

The given equation is :-  

$y=x^4-3x^2-4$.

We know that a graph is symmetrical about x-axis, if a point $\left(x,y\right)$ lies on graph, then $\left(x,-y\right)$ is also lies on graph.

So to check symmetry about x-axis, change $y$ by $-y$ in  the given equation, then we have :-

$-y=x^4-3x^2-4$.

This equation is not same as the given equation.

So we can conclude that the given equation is not symmetric about x-axis.

Step 3. To check symmetry about y-axis.

The given equation is :-  

$y=x^4-3x^2-4$.

We know that a graph is symmetrical about y-axis, if a point $\left(x,y\right)$ lies on graph, then $\left(-x,y\right)$ is also lies on graph.

So to check symmetry about y-axis, change $x$ by $-x$ in the given equation, then we have :-

$$\begin{gathered} y={\left(-x\right)}^4-3{(-x)}^2-4 \\ \Rightarrow y=x^4-3x^2-4 \end{gathered}$$

The resulting equation is same as the given equation.

So we can say that the given equation is symmetric about y-axis.

Step 4. To check symmetry about origin.

The given equation is :-  

$y=x^4-3x^2-4$.

We know that a graph is symmetrical about origin, if a point $\left(x,y\right)$ lies on graph, then $\left(-x,-y\right)$ is also lies on graph.

So to check symmetry about origin, change $x$ by $-x$ and $y$ by $-y$ in the given equation, then we have :-

$$\begin{gathered} -y={\left(-x\right)}^4-3{(-x)}^2-4 \\ \Rightarrow -y=x^4-3x^2-4 \end{gathered}$$

The resulting equation is not same as the given equation.

So we can say that the given equation is not symmetric about origin.