Find the equation of the parabola that opens downwards with vertex at (0, 0) and

Find the equation of the parabola that opens downwards with vertex at (0, 0) and passes through the point (2, -4).

Find the equation of the parabola that opens downwards with vertex at (0, 0) and passes through the point (2, -4).

Step 1: Understand that the vertex form of a parabola that opens downwards is given by the equation y = -a(x - h)^2 + k, where (h, k) is the vertex.
Step 2: Since the vertex is at (0, 0), we can substitute h = 0 and k = 0 into the equation. This simplifies our equation to y = -a(x - 0)^2 + 0, or y = -ax^2.
Step 3: We know the parabola passes through the point (2, -4). This means when x = 2, y should equal -4.
Step 4: Substitute x = 2 and y = -4 into the equation y = -ax^2. This gives us -4 = -a(2^2).
Step 5: Simplify the equation: -4 = -a(4) becomes -4 = -4a. Dividing both sides by -4 gives us 1 = a.
Step 6: Now that we have a = 1, substitute it back into the equation y = -ax^2. This gives us y = -1x^2, or simply y = -x^2.
Step 7: Therefore, the equation of the parabola is y = -2x^2.

Vertex Form of a Parabola – The vertex form of a parabola is given by y = a(x - h)² + k, where (h, k) is the vertex. For a parabola that opens downwards, 'a' must be negative.
Substituting Points – To find the specific equation of a parabola, you substitute a known point into the vertex form to solve for the coefficient 'a'.