Can someone explain what the graph would look like for :
y=-4x^2-2 and it's inverse
if you're talking about inverses, you must have done some work by now with quadratics. You know that y is a parabola, opening downward.
The graph has two values of x for each y, so there is no general inverse.
To start, let's first graph the equation y = -4x^2 - 2.
The given equation is in the form of a quadratic function. The coefficient of x^2 (-4) tells us that the graph will be a downward-opening parabola, and the constant term (-2) represents a vertical shift downward by 2 units.
Now, let's plot some points to create the graph:
When x = -2:
y = -4(-2)^2 - 2 = -4(4) - 2 = -16 - 2 = -18
So one point we have is (-2, -18).
When x = -1:
y = -4(-1)^2 - 2 = -4(1) - 2 = -4 - 2 = -6
So another point is (-1, -6).
When x = 0:
y = -4(0)^2 - 2 = -4(0) - 2 = 0 - 2 = -2
Another point is (0, -2).
When x = 1:
y = -4(1)^2 - 2 = -4(1) - 2 = -4 - 2 = -6
Another point is (1, -6).
When x = 2:
y = -4(2)^2 - 2 = -4(4) - 2 = -16 - 2 = -18
One more point is (2, -18).
We can plot these points and sketch the graph as follows:
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|--18--|
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____________
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|-2----|-6---| -----2-----6----- |
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--2--
Now, let's find the inverse of this function. To do that, we need to swap the x and y variables and solve for y.
So, we have:
x = -4y^2 - 2
Rearranging and solving for y, we get:
-4y^2 = x + 2
y^2 = -(x + 2)/4
y = ±√(-(x + 2)/4)
Since the original function is not one-to-one, its inverse will only be the part of the graph for x ≤ 0.
We can plot the inverse graph by reflecting the part of the original graph in the line y = x:
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----------------------------
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--2--
--6--
-10--
The graph of the inverse function y = √(-(x + 2)/4) is shown above, only for the part where x ≤ 0.
To understand what the graph of the equation y = -4x^2 - 2 looks like and its inverse, we can follow a few steps:
1. Graphing the equation y = -4x^2 - 2:
- Start by creating a table of x and y values. Choose various x values and calculate the corresponding y values using the equation.
- For example, when x = -2, y = -4(-2)^2 - 2 = -4(4) - 2 = -16 - 2 = -18. Repeat this process for several x values.
- The table might look something like this:
x | y
-------
-2 | -18
-1 | -6
0 | -2
1 | -6
2 | -18
- Plot the points on a graph and connect them to form a smooth curve. Since the coefficient of x^2 is negative, the graph will be an upside-down "U" shape known as a parabola. Additionally, the graph will shift downward by 2 units because of the "-2" term in the equation.
2. Finding the inverse of the function:
- To find the inverse of a function, swap the x and y variables and solve for y.
- The equation y = -4x^2 - 2 becomes x = -4y^2 - 2.
- Rearrange the equation and solve for y:
-4y^2 = x + 2
y^2 = -(x + 2)/4
y = ±√(-(x + 2)/4)
- The inverse function has two branches because of the ± sign, and it is a reflection of the original function over the line y = x (the diagonal line through the origin).
- Plot the inverse function on the same graph as the original function, swapping the x and y values from the table created earlier.