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

₹0.0

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

Options:

Correct Answer: y = -2x^2

Solution:

Using the vertex form and substituting the point (2, -4), we find that the equation is y = -2x^2.

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'.