Question

for the given equation, list the intercepts and test for symmetry.
$x^{2}+16y^{2}=16$
what are the intercept(s)? select the correct choice below and fill in any answer boxes within your choice.
a. the intercept(s) is/are \square.
(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)

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

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

Substitute $x=0$ into $x^2 + 16y^2 = 16$:
$0^2 + 16y^2 = 16$
$16y^2 = 16$
$y^2 = 1$
$y = \pm 1$
So y-intercepts are $(0, -1)$ and $(0, 1)$.

Step3: Test symmetry about x-axis

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

Step4: Test symmetry about y-axis

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

Step5: Test symmetry about origin

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

Answer:

Intercepts:

A. The intercept(s) is/are $(-4, 0), (4, 0), (0, -1), (0, 1)$

Symmetry:

The graph is symmetric about the x-axis, y-axis, and the origin.