Graph y = -4x^2 - 2 and its inverse.

To graph the function y = -4x^2 - 2, follow these steps:

1. Identify that this is a quadratic function in the form of y = ax^2 + bx + c, where a = -4, b = 0, and c = -2. 2. Since a < 0, the parabola opens downwards.

3. The vertex of the parabola can be found by using the vertex formula, h = -b/(2a) and k = c - (b^2)/(4a). Since b = 0, the vertex is at x = 0. Plugging x into the equation gives us y = -2; hence, the vertex is at (0, -2).

4. Plot the vertex (0, -2) on the coordinate plane.

5. To find additional points, choose x-values and plug them into the equation to find the corresponding y-values. For instance: - When x = 1, y = -4(1)^2 - 2 = -4 - 2 = -6. Point: (1, -6) - When x = -1, y = -4(-1)^2 - 2 = -4 - 2 = -6. Point: (-1, -6) (Choosing symmetric points around the vertex makes graphing easier since parabolas are symmetrical.)

6. Plot the points you found and draw the parabola opening downwards through these points.

Inverse of the function: Mathematically, the inverse of a function is found by swapping the x and y variables and then solving for y. However, y = -4x^2 - 2 is not a one-to-one function (a function where each x-value maps to a unique y-value), so it does not have an inverse that is a function without restricting its domain.

Nevertheless, if we were to find the inverse relation (understanding that it will not be a function unless we restrict the domain), we'd follow these steps:

1. Replace y with x and x with y: x = -4y^2 - 2

2. Solve for y: x + 2 = -4y^2 (x + 2)/-4 = y^2 y = ±√((x + 2)/-4)

3. Since taking the square root yields two solutions, we split this into two separate relations: y = √((x + 2)/-4) (which is not valid for our function because our parabola opens downward) y = -√((x + 2)/-4)

For the inverse, however, we will consider y = -√((x + 2)/-4) because it corresponds to the downward opening parabola and plot this relation to create a graph that reflects the original parabola over the line y = x.

Graphing the inverse relation accurately requires that for each point (x, y) on the parabola, we plot a point (y, x). Here, the graph will be a semi-parabola because the original was a function; points are chosen such that x = -4y^2 - 2 is valid.

Due to the complexity of graphing the inverse relation and considering it is not a function, detailed plotting is not provided here, but you can visualize it as a curve reflecting the points of the original parabola across the line y = x.

Extra: A function and its inverse essentially "undo" each other; that is, if f is a function, then f^-1(f(x)) = x. However, not all functions have inverses that are also functions. For a function to have an inverse that is a function, it must be one-to-one, meaning that each y-value is associated with exactly one x-value.

Quadratic functions, such as y = -4x^2 - 2, are not one-to-one because they fail the Horizontal Line Test; if you draw a horizontal line through the graph, it will intersect the parabola at more than one point, showing that the same y-value corresponds to multiple x-values.

Graphing the inverse relation of a function that is not one-to-one requires a little bit of caution since not all of the y-values will correspond to a single x-value. For these types of functions, it is common to restrict their domain, which allows us to consider only a portion of the parabola that passes the Horizontal Line Test, and thus, its inverse can also be considered a function over that restricted domain.