Chapter 12, Problem 9STP

To determine

To find: Which of the following equations is symmetrical about y-axis.

Answer to Problem 9STP

Option-B is correct

  $y=6x^2+9$

Explanation of Solution

If a function is symmetrical with respect to the y-axis

Then,

  $f\left(x\right)=f\left(-x\right)$

For Option-A

Given

  $y=f\left(x\right)=x^2+3x-1$

We need to find $f\left(-x\right)$ and compare with $f\left(x\right)$

  $\begin{array}{l}\Rightarrow f\left(-x\right)={\left(-x\right)}^2+3\left(-x\right)-1\\ \Rightarrow f\left(-x\right)=x^2-3x-1\end{array}$

Since, $f\left(-x\right)$ is not equal to $f\left(x\right)$

So, $y=f\left(x\right)=x^2+3x-1$ is Not symmetrical about the y-axis.

So, Option A is Wrong.

For Option-B

Given

  $y=f\left(x\right)=6x^2+9$

We need to find $f\left(-x\right)$ and compare with $f\left(x\right)$

  $\begin{array}{l}\Rightarrow f\left(-x\right)=6{\left(-x\right)}^2+9\\ \Rightarrow f\left(-x\right)=6x^2+9\\ \Rightarrow f\left(-x\right)=f\left(x\right)\end{array}$

Since, $f\left(-x\right)$ is equal to $f\left(x\right)$

So, $y=f\left(x\right)=6x^2+9$ is symmetrical about the y-axis.

So, Option B is Correct.

For Option-C

Given

  $y=f\left(x\right)=-x^2+x$

We need to find $f\left(-x\right)$ and compare with $f\left(x\right)$

  $\begin{array}{l}\Rightarrow f\left(-x\right)=-{\left(-x\right)}^2+\left(-x\right)\\ \Rightarrow f\left(-x\right)=-x^2-x\end{array}$

Since, $f\left(-x\right)$ is not equal to $f\left(x\right)$

So, $y=f\left(x\right)=-x^2+x$ is not symmetrical about the y-axis.

So, Option C is Wrong.

For Option-D

Given

  $y=f\left(x\right)=3x^2-3x+1$

We need to find $f\left(-x\right)$ and compare with $f\left(x\right)$

  $\begin{array}{l}\Rightarrow f\left(-x\right)=3{\left(-x\right)}^2-3\left(-x\right)+1\\ \Rightarrow f\left(-x\right)=3x^2+3x+1\end{array}$

Since, $f\left(-x\right)$ is not equal to $f\left(x\right)$

So, $y=f\left(x\right)=3x^2-3x+1$ is not symmetrical about the y-axis.

So, Option D is Wrong.