Question

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 \boxed{}.
(type an ordered pair. use a comma to separate answers as needed.)
b. there are no intercepts.

Explanation:

Step1: Find x-intercept (set y=0)

Set $y=0$, solve for $x$:
$$0^2 = x + 64$$
$$x = -64$$
So x-intercept is $(-64, 0)$.

Step2: Find y-intercept (set x=0)

Set $x=0$, solve for $y$:
$$y^2 = 0 + 64$$
$$y^2 = 64$$
$$y = \pm 8$$
So y-intercepts are $(0, 8)$ and $(0, -8)$.

Step3: Test symmetry about x-axis

Replace $y$ with $-y$:
$$(-y)^2 = x + 64$$
$$y^2 = x + 64$$
This matches the original equation, so symmetric about x-axis.

Step4: Test symmetry about y-axis

Replace $x$ with $-x$:
$$y^2 = -x + 64$$
This does not match the original equation, so no symmetry about y-axis.

Step5: Test symmetry about origin

Replace $x$ with $-x$ and $y$ with $-y$:
$$(-y)^2 = -x + 64$$
$$y^2 = -x + 64$$
This does not match the original equation, so no symmetry about origin.

Answer:

The intercept(s) is/are $(-64, 0)$, $(0, 8)$, $(0, -8)$
The graph is symmetric about the x-axis.