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 that opens downward is y = -ax^2, where 'a' is a positive constant.

Step-by-step solution

Identify Vertex Coordinates

We are given that the vertex of the parabola is at the origin. Therefore, the vertex coordinates are (0,0).

Determine Parabola Direction

Since the parabola opens downward, the coefficient 'a' in the standard equation y = ax^2 + bx + c will be negative.

Substitute Vertex Coordinates

Knowing the vertex coordinates, we can substitute them into the standard equation. So, with the vertex coordinates (0,0), we have 0 = a(0)^2 + b(0) + c.

Simplify the Equation

By simplifying the equation 0 = a(0)^2 + b(0) + c, we get 0 = c. So, the equation of the parabola now becomes y = ax^2 + bx.

Consider Vertex Form

The vertex form of a parabola equation is y = a(x - h)^2 + k, where (h, k) are the coordinates of the vertex. Since the vertex is at the origin, our equation becomes y = ax^2.

Combine Information

We know that 'a' must be negative, because the parabola opens downwards. Also, since the vertex is at the origin, the equation of the parabola is y = ax^2.

Final Answer

Combining the given information, the equation of the standard parabola with its vertex at the origin that opens downward is y = -ax^2, where 'a' is a positive constant.

Key concepts

Parabola Vertex

The vertex of a parabola is one of the most important features it has. It represents the highest or lowest point on the curve, depending on which way the parabola opens. In the case of the standard parabola equation, when the vertex is at the origin, it means that the coordinates of this pivotal point are at (0,0). This simplifies the equation significantly.

Considering our specific exercise, since we know the vertex is at the origin, any term in the equation that involves the vertex coordinates vanishes. As the vertex coordinates are (0,0), if we plug them into any form of the parabola equation, the terms with 'h' and 'k' (which represent the vertex coordinates in the vertex form of a parabola equation) will be zero. Thus, the equation simplifies to a function of 'x' only.

The understanding of the vertex provides a strong foundation for graphing and solving equations involving parabolas and can also give insights into the behavior of the function across its domain.

Direction of Parabola Opening

The direction in which a parabola opens is determined by the coefficient of the quadratic term in its equation. For standard parabolas, this is the 'a' in the equation \( y = ax^2 \). If 'a' is positive, the parabola opens upwards, and if 'a' is negative, the parabola opens downward.

In our exercise, we've determined that since the parabola opens downward, 'a' must be a negative number. This piece of information is crucial as it tells us about the behavior of the parabola and affects the set of real-world problems that this function can represent, such as the path of a projectile, the shape of a cable of a suspension bridge, or even the profit/loss potential in economics.

Understanding the impact of the sign of 'a' on the parabola's opening direction is a key concept in the study of quadratics, and mastering this will allow students to predict the shape of the graph without plotting it.

Vertex Form of a Parabola

The vertex form of a parabola is particularly useful because it readily shows the vertex of the parabola and makes graphing simpler. It is written as \( y = a(x - h)^2 + k \), where 'a' affects the width and direction of the opening, and (h, k) are the coordinates of the vertex.

In our example, since we know the vertex is at the origin (0,0) and the parabola opens downward, the vertex form becomes \( y = a(x - 0)^2 + 0 \) which reduces to \( y = ax^2 \). Because the parabola opens downwards, we know that 'a' is negative, so we have \( y = -ax^2 \) where 'a' is a positive constant.

This form is tremendously powerful when dealing with quadratic functions because it allows for easy transformations such as translations and reflections, simply by manipulating the parameters in the equation. Recognizing and working with the vertex form is a fundamental skill that can greatly enhance a student's ability to tackle a wide array of problems involving parabolas.