Question

for the given equation, list the intercepts and test for symmetry.
y = x² - x - 72

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 y-intercept (set x=0)

Substitute $x=0$ into $y=x^2 - x - 72$:
$y=0^2 - 0 - 72 = -72$
Y-intercept: $(0, -72)$

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

Solve $x^2 - x - 72 = 0$. Factor the quadratic:
$(x - 9)(x + 8) = 0$
Set each factor to 0:
$x - 9 = 0 \implies x=9$; $x + 8 = 0 \implies x=-8$
X-intercepts: $(9, 0), (-8, 0)$

Step3: Test symmetry about x-axis

Replace $y$ with $-y$: $-y=x^2 - x - 72 \implies y=-x^2 + x + 72$. This is not equivalent to original equation, so no x-axis symmetry.

Step4: Test symmetry about y-axis

Replace $x$ with $-x$: $y=(-x)^2 - (-x) - 72 = x^2 + x - 72$. This is not equivalent to original equation, so no y-axis symmetry.

Step5: Test symmetry about origin

Replace $x$ with $-x$ and $y$ with $-y$: $-y=(-x)^2 - (-x) - 72 \implies -y=x^2 + x - 72 \implies y=-x^2 - x + 72$. This is not equivalent to original equation, so no origin symmetry.

Answer:

Intercepts: $(0, -72), (9, 0), (-8, 0)$
The graph has no symmetry about the x-axis, y-axis, or origin.