1 − If a graph represented by f(x,y)=0 is symmetric with respect to x-axis, we should have f(x,y)=f(x,−y).
Here, in 6x2=y−1, we have f(x,y)=6x2−y+1=0 and f(x,−y)=6x2−(−y)+1=6x2+y+1 and hence
f(x,y)≠f(x,−y) and hence it is **not symmetric w.r.t. x-axis.
2 − If a graph represented by f(x,y)=0 is symmetric with respect to y-axis, we should have f(x,y)=f(−x,y).
In 6x2=y−1, we have f(−x,y)=6(−x)2−y+1=6x2−y+1 and hence
f(x,y)=f(−x,y) and hence it is symmetric w.r.t. y-axis.
3 − If a graph represented by f(x,y)=0 is symmetric with respect to line y=x, we should have f(x,y)=f(y,x).
In 6x2=y−1, we have f(y,x)=6y2−x+1 and hence
f(x,y)≠f(y,x) and hence it is not symmetric w.r.t. line y=x.
4 − If a graph represented by f(x,y)=0 is symmetric with respect to line y=−x, we should have f(x,y)=f(−y,−x).
In 6x2=y−1, we have f(−y,−x)=6(−y)2−(−x)+1=6y2+x+1 and hence
f(x,y)≠f(−y,−x) and hence it is not symmetric w.r.t. line y=−x.
graph{(6x^2-y+1)(x-y)(x+y)=0 [-5.394, 4.606, -0.64, 4.36]}
