Question

The graph of x^3+y^3=4 is symmetric with respect to which of the following?

a.
y-axis
c.
line y = x
b.
x-axis
d.
none of the above

Answer 1

Option C is the correct option.

Explanation:

The given graph is in the form of x³+y³ = 4

To check the symmetry of the function about y-axis we put (-x) in place of x in the given function and if we get no change in the function then it is symmetric to y- axis.

(-x)³ + y³ = 4

-x³ + y³ = 4

This function is different from the original one. Therefore non symmetric to the y axis.

Similarly for x axis we put (-y) in place of y the equation becomes as

x³ + (-y)³ = 4

x³ - y³ = 4

Again the given function is non symmetric to x axis

for y = x

by putting x in place of y and y in place of x we get

y³ + x³ = 4

Which exactly same as the original equation. therefore the function is symmetric to the line y = x.

Option C is the correct answer.

Answer 2

Choice C. The line y =x

Explanation:

A function f(x) is symmetric with respect to the y-axis if f(-x) = f(x). If we substitute -x in place of x we end up with a different function. So the function is not symmetric with respect to the y-axis.

Moreover, the function is not symmetric with respect to the x-axis since substituting -y in place of y results into a new function.

If we plot the graph of the function together with the line y =x on the same graph we note that the line y =x acts as a mirror line such that one half of the function lies on the upper end and the other half on the lower end. The function is thus symmetric with respect to the line y =x.