Question

for the given equation, list the intercepts and test for symmetry.
$x^2 + y - 16 = 0$
what are the intercept(s)? select the correct choice below and fill in any answer boxes within your choice.
a.
(type an ordered pair. use a comma to separate answers as needed.)
b. there are no intercepts.

Explanation:

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

Set $y=0$ in $x^2 + y - 16 = 0$:
$x^2 + 0 - 16 = 0$
$x^2 = 16$
$x = \pm 4$
So x-intercepts are $(-4, 0)$ and $(4, 0)$.

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

Set $x=0$ in $x^2 + y - 16 = 0$:
$0^2 + y - 16 = 0$
$y = 16$
So y-intercept is $(0, 16)$.

Step3: Test symmetry about y-axis

Replace $x$ with $-x$:
$(-x)^2 + y - 16 = x^2 + y - 16 = 0$, which matches original equation. Symmetric about y-axis.

Step4: Test symmetry about x-axis

Replace $y$ with $-y$:
$x^2 - y - 16 = 0$, which does not match original equation. Not symmetric about x-axis.

Step5: Test symmetry about origin

Replace $x$ with $-x$ and $y$ with $-y$:
$(-x)^2 - y - 16 = x^2 - y - 16 = 0$, which does not match original equation. Not symmetric about origin.

Answer:

Intercepts:

A. $(-4, 0), (4, 0), (0, 16)$

Symmetry:

The graph is symmetric about the y-axis; it has no symmetry about the x-axis or the origin.