Hi, i hadd a problem y= ((8)/(x^3))-((6)/(x))
I need to know hoe the graph is symetric with respect to the
1)x axis
2)y axis
3)origin
I just needed to know the analytical way to find it without looking at the graph
The graph of a function is symmetric with respect to:
- the y-axis if f(x) = f(-x) for all $\displaystyle x \in domain$
- the origin if f(x) = -f(-x) for all $\displaystyle x \in domain$
If a graph is symmetric to the x-axis it is not the graph of a function:
For every x-value you have pairs of y-values such that $\displaystyle y_1 = -y_2$
To your function: $\displaystyle f(x)=\dfrac8{x^3} - \dfrac6x = \dfrac{8-6x^2}{x^3}$
a) Symmetry to the y-axis:
$\displaystyle \dfrac{8-6x^2}{x^3} = \dfrac{8-6(-x)^2}{(-x)^3}$
$\displaystyle \dfrac{8-6x^2}{x^3} \neq -\dfrac{8-6(x)^2}{(x)^3}$
That means: No symmetry to the y-axis.
b) Symmetry to the origin:
$\displaystyle \dfrac{8-6x^2}{x^3} = -\dfrac{8-6(-x)^2}{(-x)^3}$
$\displaystyle \dfrac{8-6x^2}{x^3} = \dfrac{8-6(x)^2}{(x)^3}$
That means: Symmetry to the origin.