Chapter 3.3, Problem 2MCQ

To determine

To find: whether graph of equation is symmetric with respect to x-axis, y-axis, the line $y=x$ , the line $y=-x$ , origin or none of these.

Answer to Problem 2MCQ

None

Explanation of Solution

Given information:

  $5x^2+6x-9=y$

Calculation:

Put $\left(a,b\right)$ into the equation and solve as follows:

  $\begin{array}{l}5x^2+6x-9=y\\ 5a^2+6a-9=b\end{array}$

If the solution of equation is same as that of any other case, then their graphs are symmetric.

Let’s check whether the given cases are symmetric with the given equation or not by substituting replacing $\left(x,y\right)$with different coordinates in the equation and simplify as shown below.

Check symmetry with	$\left(x,y\right)$	$5x^2+6x-9=y$	Symmetric with graph equation (Yes/No)
x-axis	$\left(a,-b\right)$	$5a^2+6a-9=-b$	No
y-axis	$\left(-a,b\right)$	$\begin{array}{c}5{\left(-a\right)}^2+6\left(-a\right)-9=b\\ 5a^2-6a-9=b\end{array}$	No
$y=x$	$\left(b,a\right)$	$5b^2+6b-9=a$	No
$y=-x$	$\left(-b,-a\right)$	$\begin{array}{c}5{\left(-b\right)}^2+6\left(-b\right)-9=-a\\ 5b^2-6b-9=-a\end{array}$	No

Let’s check whether the graph is symmetric to origin or not. First convert the equation in $f\left(x\right)$ as follows:

  $\begin{array}{c}y=5x^2+6x-9\\ f\left(x\right)=5x^2+6x-9\left\{\text{In terms of }f\left(x\right)\right\}\end{array}$

Compare the values of $f\left(-x\right)$ and $-f\left(x\right)$ as shown below.

$f\left(x\right)=5x^2+6x-9$
$\begin{array}{c}f\left(-x\right)=5{\left(-x\right)}^2+6\left(-x\right)-9\\ =5x^2-6x-9\end{array}$	$\begin{array}{c}-f\left(x\right)=-\left(5x^2+6x-9\right)\\ =-5x^2-6x+9\end{array}$

Since, $f\left(-x\right)$ is not equal to $-f\left(x\right)$ . Thus the graph is not symmetric to the origin.