Problem 54 - Algebra 2

Question

OPEN ENDED Write the equation of a parabola with a vertex of \((2,-1)\) and which opens downward.

Step-by-Step Solution

Answer

The equation is \(y = -(x - 2)^2 - 1\).

Step 1: Understanding the Vertex Form

The equation of a parabola in vertex form is given by \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola. Here, we are provided with the vertex \((2, -1)\), so the equation becomes \(y = a(x - 2)^2 - 1\).

Step 2: Determining the Direction of Opening

The parabola opens downward, which means the value of \(a\) must be negative. This gives us \(y = -a(x - 2)^2 - 1\). Generally, we can pick \(a = -1\) for simplicity unless additional conditions state otherwise, resulting in the equation \(y = -(x - 2)^2 - 1\).

Step 3: Writing the Final Equation

Now we substitute \(a = -1\) into the vertex form equation. Therefore, the equation of the parabola is \(y = -(x - 2)^2 - 1\).

Key Concepts

Vertex Form

The vertex form of a quadratic equation is a powerful way to express parabolas. This is especially true when you know the vertex of the parabola.
The vertex form is written as follows:

• \( y = a(x - h)^2 + k \)

Here,

• \((h, k)\) represents the vertex of the parabola.
• The constant \(a\) affects the width and direction of the parabola.

Using the vertex form makes it easy to identify and utilize the vertex, streamlining the process of writing a parabola's equation given a vertex.

Direction of Opening

The direction in which a parabola opens is determined by the value of the coefficient \(a\) in the vertex form equation. If \(a > 0\), the parabola opens upwards, similar to a smiling face. However, if \(a < 0\), the parabola opens downward, just like a frowning face.
In the exercise, since the parabola opens downward, we conclude \(a\) must be negative.

• This knowledge is crucial as it impacts the shape of the parabola and whether the vertex represents a maximum or minimum point.

When you encounter problems involving parabolas, checking the sign of \(a\) will immediately inform you about the direction of opening, providing a clear insight into the parabola's overall behavior.

Vertex

In parabolas written in vertex form, the vertex \((h, k)\) is the point where the parabola changes direction. It's essentially the highest or lowest point on a parabola. For upwards opening parabolas, it is the minimum point, while for downwards opening parabolas, it's the maximum.
The vertex offers a wealth of information about the parabola:

• It directly tells us where the parabola is positioned on the graph.
• It indicates the axis of symmetry, a vertical line that passes through \(h\).

Learning how to extract and use vertex information is crucial in graphing and understanding quadratic equations, as it allows you to visualize the parabola's shape and key features.

Quadratic Equation

The term "quadratic equation" encompasses expressions like parabolas written in different forms, including standard and vertex forms. Quadratic equations are polynomials of the second degree, with the general form being:

• \( ax^2 + bx + c = 0 \)

In vertex form, it specifically highlights the vertex, making it easier to identify certain characteristics like the vertex's position and direction of opening.
This equation type commonly models various scenarios in physics, engineering, and economics. Understanding quadratic equations helps in solving, graphing, and applying mathematical models to real-world problems.
Recognizing the differences and transformations between forms provides a solid foundation for tackling quadratic problems with ease.