find any intercepts of the graph of the given equation. determine whether the graph of the equation possesses symmetry with respect to the x-axis, y-axis, or orgin

y=x^2-2x-2

y = (x-1)^2 -3

When y = 0, (x-1)^2 = 3
x - 1 = + or - sqrt3
x = 1 + sqrt3 or 1 - sqrt3

When x = 0, y = -3

There is no symmetry about the x or y axes, or origin.

When x = 0, y = -2, y axis intercept

When y = 0
x = [ 2 +/- sqrt(4+8)]/2 = 1 +/- sqrt 3

therefore parabola is symmetric about the vertical line x = 1

show it by completing the square

x^2 - 2x = y + 2

x^2 - 2x + 1 = y+3

(x-1)^2 = (y+3)
vertex at (1,-3)

thank you damon! that was exactly what i was looking for, by the way it's 4:32 am here and i've been searching for that answer for hours...

To find the intercepts of the graph of the given equation, we need to determine the coordinates where the graph intersects the x-axis and the y-axis.

1. X-Intercepts: These are the points where the graph intersects the x-axis, meaning the value of y is 0. To find the x-intercepts, we set y = 0 and solve for x.

y = x^2 - 2x - 2
0 = x^2 - 2x - 2

To solve this quadratic equation, we can factor it or use the quadratic formula.

Using factoring:
0 = (x - 2)(x + 1)
Setting each factor to 0 gives us two solutions: x - 2 = 0 or x + 1 = 0
Solving these equations, we find x = 2 and x = -1 are the x-intercepts.

Therefore, the graph intersects the x-axis at the points (2, 0) and (-1, 0).

2. Y-Intercept: This is the point where the graph intersects the y-axis, meaning the value of x is 0. To find the y-intercept, we set x = 0 and solve for y.

y = (0)^2 - 2(0) - 2
y = -2

Therefore, the graph intersects the y-axis at the point (0, -2).

Now let's determine the symmetry of the graph with respect to the x-axis, y-axis, or origin.

Symmetry with respect to the x-axis: If replacing y by -y in the equation yields an equivalent equation, then the graph is symmetric with respect to the x-axis.

Let's substitute -y for y and simplify the equation:
-y = x^2 - 2x - 2

Using properties of equality, we can multiply both sides by -1:
y = -x^2 + 2x + 2

The equation is not equivalent to the original equation, so the graph does not possess symmetry with respect to the x-axis.

Symmetry with respect to the y-axis: If replacing x by -x in the equation yields an equivalent equation, then the graph is symmetric with respect to the y-axis.

Let's substitute -x for x and simplify the equation:
y = (-x)^2 - 2(-x) - 2
y = x^2 + 2x - 2

The equation is not equivalent to the original equation, so the graph does not possess symmetry with respect to the y-axis.

Symmetry with respect to the origin: If replacing x by -x and y by -y in the equation yields an equivalent equation, then the graph is symmetric with respect to the origin.

Let's substitute -x for x and -y for y:
-y = (-x)^2 - 2(-x) - 2
y = x^2 + 2x - 2

The equation is the same as the one obtained when checking for symmetry with respect to the y-axis. So, the graph does not possess symmetry with respect to the origin.

In summary, the graph intersects the x-axis at (2, 0) and (-1, 0), the y-axis at (0, -2), and does not possess symmetry with respect to the x-axis, y-axis, or origin.