Questions: (b) (x=y^3+3 y) Make a table of values and sketch the graph of the equation. Find the (x) - and (y)-intercepts. (If an answer does not exist, enter DNE.) (x)-intercept: ((x, y)=(square)) (y)-intercept: ((x, y)=(square)) Test for symmetry. (Select all that apply.)

Solution Steps

The x-intercept is the point where the graph crosses the x-axis, which means y = 0. Substituting y = 0 into the equation x = y³ + 3y gives x = (0)³ + 3(0) = 0. Therefore, the x-intercept is (0, 0).

The y-intercept is the point where the graph crosses the y-axis, which means x = 0. Substituting x = 0 into the equation gives 0 = y³ + 3y, or y(y² + 3) = 0. This equation is satisfied when y = 0. Since y² + 3 will always be positive for real values of y, the only solution is y=0. Therefore, the y-intercept is (0, 0).

• Symmetry with respect to the x-axis: Replace y with -y in the equation. The new equation is x = (-y)³ + 3(-y) = -y³ - 3y, which is not the same as the original equation. Thus, the graph is not symmetric with respect to the x-axis.
• Symmetry with respect to the y-axis: Replace x with -x in the equation. The new equation is -x = y³ + 3y, which is not the same as the original equation. Thus, the graph is not symmetric with respect to the y-axis.
• Symmetry with respect to the origin: Replace x with -x and y with -y in the equation. The new equation is -x = (-y)³ + 3(-y) = -y³ - 3y, which simplifies to x = y³ + 3y. This is the same as the original equation. Therefore, the graph is symmetric with respect to the origin.

Final Answer