List the intercepts and type(s) of symmetry, if any.
-$$y = \frac { - x ^ { 5 } } { x ^ { 2 } - 6 }$$
A) intercept: $( 0,0 )$
symmetric with respect to origin

B) intercept: $( 0,0 )$
symmetric with respect to $x$-axis

C) intercepts: $( \sqrt { 6 } , 0 ) , ( - \sqrt { 6 } , 0 ) , ( 0,0 )$
symmetric with respect to origin

D) intercept: $( 0,0 )$
symmetric with respect to $\mathrm { y }$-axis

Question

List the intercepts and type(s) of symmetry, if any.
-$$y = \frac { - x ^ { 5 } } { x ^ { 2 } - 6 }$$
A) intercept: $( 0,0 )$
symmetric with respect to origin

B) intercept: $( 0,0 )$
symmetric with respect to $x$-axis

C) intercepts: $( \sqrt { 6 } , 0 ) , ( - \sqrt { 6 } , 0 ) , ( 0,0 )$
symmetric with respect to origin

D) intercept: $( 0,0 )$
symmetric with respect to $\mathrm { y }$-axis

Quizplus · Accepted Answer

The given function is $y = \frac{-x^5}{x^2 - 6}$. To find the intercepts, we set $x = 0$ to find the y-intercept and $y = 0$ to find the x-intercepts. Setting $x = 0$, we get $y = 0$, indicating an intercept at $(0,0)$. Setting $y = 0$, we find that the numerator must be zero, which happens when $x = 0$, confirming the intercept at $(0,0)$ again. There are no other x-intercepts because for $y = 0$, the equation simplifies to $0 = -x^5$, which only has a solution at $x = 0$. The function is odd, as replacing $x$ with $-x$ results in the original function being negated ($-y$), indicating symmetry with respect to the origin.

List the Intercepts and Type(s) of Symmetry, If Any y=−x5x2−6y = \frac { - x ^ { 5 } } { x ^ { 2 } - 6 }y=x2−6−x5​

List the Intercepts and Type(s) of Symmetry, If Any $y = \frac { - x ^ { 5 } } { x ^ { 2 } - 6 }$