Problem

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.

Answer

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Answer

\(\boxed{\text{The y-intercept(s) is/are (-8, 0) and (8, 0). The x-intercept is (0, -64). The graph is symmetric with respect to the x-axis.}}\)

Steps

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 :\(\boxed{\text{The y-intercept(s) is/are (-8, 0) and (8, 0). The x-intercept is (0, -64). The graph is symmetric with respect to the x-axis.}}\)

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