graph the function using transformations, show and state , a) the domain, B) range c) the asymptotes

F(x)= 1
----- -4
x^2


please show work!

To graph the function F(x), we'll start with the parent function f(x) = 1/x^2 and apply transformations.

a) The domain of the function F(x) is all real numbers except x = 0, because division by zero is undefined.

b) To find the range of the function F(x), we need to study the behavior of the parent function f(x) = 1/x^2. Since f(x) is a reciprocal function, its range does not include zero. However, it can take any positive or negative value.

c) Asymptotes:
- Vertical asymptote: The vertical asymptote occurs when the denominator of the function F(x) equals zero, which is x^2 = 0. Solving for x, we find x = 0. Therefore, x = 0 is a vertical asymptote.
- Horizontal asymptote: The horizontal asymptote can be determined by looking at the behavior of the function as x approaches positive or negative infinity. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.

Now, let's apply the transformations to graph the function F(x):

1. Vertical shift: Since there are no additions or subtractions outside the function, there is no vertical shift.
2. Horizontal shift: Since there are no additions or subtractions inside the function, there is no horizontal shift.
3. Vertical stretch or compression: Since there is a division by x^2, the graph is vertically compressed.
The compression is determined by taking the reciprocal of the constant: 1/(-4) = (-1/4). This means the graph is compressed vertically by a factor of 4.
4. Vertical reflection: Since there is a negative sign outside the function, the graph is vertically reflected.

Since the parent function f(x) is a simple reciprocal curve with a horizontal asymptote at y = 0 and no zeroes, we know it looks like an upside-down U-shape.
After applying the transformations, the graph of F(x) will have the same general shape but compressed and reflected.

To graph F(x), start by plotting a few key points on the parent function f(x) = 1/x^2 , and then apply the transformations:

For f(x):
- When x = -2, f(x) = 1/(-2)^2 = 1/4, so one point is (-2, 1/4).
- When x = -1, f(x) = 1/(-1)^2 = 1, so another point is (-1, 1).
- When x = 1, f(x) = 1/(1)^2 = 1, so another point is (1, 1).
- When x = 2, f(x) = 1/(2)^2 = 1/4, so another point is (2, 1/4).

Now, apply the transformations:
- Vertical compression: Multiply the y-values (vertical coordinates) of the key points by -4.
The transformed points become: (-2, -1), (-1, -4), (1, -4), (2, -1).
- Vertical reflection: Multiply these transformed y-values by (-1).
The final transformed points become: (-2, 1), (-1, 4), (1, 4), (2, 1).

Using these transformed points, plot them on the coordinate plane. Connect the points smoothly with a curve considering the general shape.

Finally, label the vertical asymptote x = 0 and the horizontal asymptote y = 0.

This is the graph of the function F(x) = 1/(-4x^2).