List the intercepts and test for symmetry. \[ y^{2}=x+64 \] Select the correct choice below and fill in any answer boxes within your choice. A. The intercept(s) is/are (Type an ordered pair. Use a comma to separate answers as needed.) B. There are no intercepts.
Solution
Step 1 ：Set x=0 in the equation to find the y-intercept(s).
Step 2 ：Solving gives y = -8 and y = 8, so the y-intercepts are (-8, 0) and (8, 0).
Step 3 ：Set y=0 in the equation to find the x-intercept.
Step 4 ：Solving gives x = -64, so the x-intercept is (0, -64).
Step 5 ：Test for symmetry with respect to the y-axis by replacing x with -x in the equation. The equation does not remain the same, so the graph is not symmetric with respect to the y-axis.
Step 6 ：Test for symmetry with respect to the x-axis by replacing y with -y in the equation. The equation remains the same, so the graph is symmetric with respect to the x-axis.
Step 7 ：Test for symmetry with respect to the origin by replacing (x, y) with (-x, -y) in the equation. The equation does not remain the same, so the graph is not symmetric with respect to the origin.
Step 8 ：\(\boxed{\text{The y-intercept(s) is/are (-8, 0) and (8, 0). The x-intercept is (0, -64). The graph is symmetric with respect to the x-axis.}}\)
