Question

What is the equation of the standard parabola with its vertex at the origin that opens downward?

Short answer

Answer: The equation of the standard parabola with its vertex at the origin and opening downward is \(y = -x^2\).

Step-by-step solution

Recall the general equation of a parabola

There are two forms of the general equation of a parabola, one for parabolas opening vertically (upward or downward) and another for parabolas opening horizontally (left or right): - Vertical (opens upward/downward): \(y = a(x - h)^2 + k\) - Horizontal (opens left/right): \(x = a(y - k)^2 + h\) In our case, the parabola opens downward, so we will use the vertical parabola equation.

Determine the vertex

We are given that the parabola has its vertex at the origin (0,0). This means that the h and k values in our equation will be 0: Vertex: (h, k) = (0, 0)

Determine the value of a based on the direction of the parabola

To determine in which direction the parabola opens, we need to look at the value of the coefficient 'a' in our equation. In the vertical equation, 'a' determines whether the parabola opens upward or downward: - If 'a' is positive, the parabola opens upward. - If 'a' is negative, the parabola opens downward. Since we are given that our parabola opens downward, we know that the value of 'a' will be negative. It's enough to ensure that 'a' is negative, any negative value will do. Let's choose a = -1 for simplicity.

Write the equation of the parabola

Now that we have values for a, h, and k, we can write the equation of our downward parabola with the vertex at the origin: \(y = -1(x - 0)^2 + 0\) Simplifying, we get the equation of the parabola: \(y = -x^2\) And this is the equation of the standard parabola with its vertex at the origin that opens downward.