Problem 65 - Algebra 2

Question

Which equation represents a parabola that opens downward? \(F x=-2 v^{2}\) G. \(x=2 y^{2}\) H. \(y=-2 x^{2}\) \ J. \(y=2 x^{2}\)

Step-by-Step Solution

Answer

The equation that represents a parabola opening downward is H: \(y=-2 x^{2}\)

Step 1: Identify the equations with square terms

Looking at our options, we see that all of them have either \(y^{2}\) or \(x^{2}\), which are the usual forms a parabolic equation can take. We have \(F: x=-2 v^{2}\), \(G: x=2 y^{2}\), \(H: y=-2 x^{2}\), and \(J: y=2 x^{2}\).

Step 2: Look for negative coefficients

As per our initial analysis, for a parabola to open downward, the coefficient of the square term must be negative. This occurs in options F: \(x=-2 v^{2}\) and H: \(y=-2 x^{2}\). The other two options G: \(x=2 y^{2}\) and J: \(y=2 x^{2}\) have positive coefficients of 2 and hence, the parabolas they represent open upwards, so we can exclude them.

Step 3: Consider the variable that varies

In option F: \(x=-2 v^{2}\), x varies as a function of v, so the parabola would be horizontal, either opening on the left (if the coefficient of the square term is positive) or on the right (if it’s negative). In option H: \(y=-2 x^{2}\), y varies as a function of x, so the parabola would be vertical: opening upwards if the coefficient of the square term is positive and downwards if it’s negative.

Step 4: Determine the final answer

After analyzing each option, we find that equation H: \(y=-2 x^{2}\) represents a vertical parabola that opens downward due to its negative coefficient.

Key Concepts

Equations of Parabolas

In mathematics, a parabola is a U-shaped curve that can open either upward or downward. The standard equation for a parabola is typically written as either

• Vertical: \(y = ax^2 + bx + c\)
• Horizontal: \(x = ay^2 + by + c\)

The key component here is the square term, such as \(x^2\) or \(y^2\), which defines the nature of the curve. When we evaluate different equations involving parabolas, we pay close attention to these square terms and their coefficients. If you know the orientation of a parabola (vertical or horizontal), it can help determine how the curve looks and behaves. The coefficient in front of the square term is crucial, as it affects both the direction and the width of the parabola.

Direction of Parabolas

The direction in which a parabola opens is determined by the sign and position of the coefficients associated with the square term. Generally:

• For a vertical parabola (e.g., \(y = ax^2\)), if \(a > 0\), the parabola opens upward, while if \(a < 0\), it opens downward.
• For a horizontal parabola (e.g., \(x = ay^2\)), if \(a > 0\), the parabola opens to the right; if \(a < 0\), it opens to the left.

Let's look at the example of \(y = -2x^2\) from our exercise. Here, the coefficient of \(x^2\) is negative, so this parabola opens downward. Understanding the role of the coefficient helps you quickly determine orientation and can guide graphing and real-life applications.

Quadratic Functions

Quadratic functions form the backbone of parabolic equations. They are typically expressed in the form:

• \(y = ax^2 + bx + c\)

This structure allows you to describe many natural phenomena, from projectile motion to optimizing business profits. A quadratic function yields a parabola when graphed. The vertex of this parabola, which can be found using vertex formulas, indicates the maximum or minimum point of the function depending on its orientation. For example, with \(y = -2x^2\), the negative coefficient means that the vertex is a maximum point. Quadratic functions are integral in calculus, physics, and economics due to their ability to model real-world scenarios with parabolic shapes.