Question

The graph of a relation is
(i) Symmetric with respect to the $x$-axis provided that whenever (a, b) is a point on the graph, so is $( a ,- b )$
(ii) Symmetric with respect to the $y$-axis provided that whenever $(a, b)$ is a point on the graph, so is $(- a , b )$
(iii) Symmetric with respect to the origin provided that whenever (a, b) is a point on the graph, so is $(- a ,- b )$
(iv) Symmetric with respect to the line $y = x$, provided that whenever $( a , b )$ is a point on the graph, so is $(b, a)$
The graph of the relation $x^4+y^3=1$ is symmetric with respect to

• A. the $x$-axis
• B. the $y$-axis
• C. the origin
• D. the line $y=x$

Answer 1

Answer: (B)

(i) If $x^4+y^3=1$, then we know $(-x, y)$ is on the graph since $(-x)^4+y^3=x^4+y^3=1$. And in general, when the coordinate is raised to an even power every single time in the equation, then symmetry by the other axis occurs. Since $y$ is raised to an odd number, then $x$-axis and origin symmetry are ruled out. Symmetry about $y=x$ is another story, but since $x^4+y^3=x^3+y^4$ is not necessarily true, it is ruled out. The only symmetry is about the $y$-axis.