Answer:
The equation of a parabola that contains the point (2, ‒12) and is congruent to the given parabola y = 3x^2 is y = 3(x-2)^2 - 12.
The equation of a parabola that contains the point (0, 8) and is congruent to the given parabola y = 3x^2 is y = 3x^2 + 8.
The equation of a parabola similar to the given parabola y = 3x^2 that does not contain the point (0, 0) but does contain the point (2, 2) is y = 3/4 x^2 + 2.
Explanation:
Given: y = 3x^2
To find the equation of a parabola that contains the point (2, -12) and is congruent to the given parabola, we can start by considering the basic form of a parabola, y = a(x-h)^2 + k. The point (h, k) represents the vertex of the parabola.
Since the given parabola has its vertex at (0, 0), we can start by translating the vertex to (2, -12) using the transformation (x-h) and (y-k):
Translation of the vertex: (x-2) and (y+12)
Applying this transformation to the given parabola equation, we have:
y = 3(x-2)^2 + (-12)
Simplifying the equation, we get:
y = 3(x^2 - 4x + 4) - 12
y = 3x^2 - 12x + 12 - 12
y = 3x^2 - 12x
Therefore, the equation of the parabola that contains the point (2, -12) and is congruent to the given parabola y = 3x^2 is y = 3x^2 - 12x.
To find the equation of a parabola that contains the point (0, 8) and is congruent to the given parabola, we can follow a similar process as above:
Translation of the vertex: (x-0) and (y-8)
Applying this transformation to the given parabola equation, we have:
y = 3(x-0)^2 + (8)
Simplifying the equation, we get:
y = 3x^2 + 8
Therefore, the equation of the parabola that contains the point (0, 8) and is congruent to the given parabola y = 3x^2 is y = 3x^2 + 8.
To find a parabola that is similar (not congruent) to the given parabola and contains the point (2, 2), we can adjust the coefficient 'a' to change the steepness of the parabola while preserving the vertex.
Let's choose 'a' to be different from 3. For example, let's set a = 2:
y = 2x^2
This equation represents a parabola similar to the given parabola y = 3x^2. However, this parabola does not contain the point (0, 0).
To ensure that it contains the point (2, 2), we can introduce a translation:
Translation of the vertex: (x-2) and (y-2)
Applying this transformation to the equation, we have:
y = 2(x-2)^2 + (2)
Simplifying the equation, we get:
y = 2(x^2 - 4x + 4) + 2
y = 2x^2 - 8x + 8 + 2
y = 2x^2 - 8x + 10
Therefore, the equation of a parabola similar to the given parabola y = 3x^2 that does not contain the point (0, 0) but does contain the point (2, 2) is y = 2x^2 - 8x + 10.