Problem 60 - A Graphical Approach to College Algebra

Question

Each equation defines a parabola. Without actually graphing, match the equation in Column I with its description in Column II. A. Vertex \((2,-4) ;\) opens downward B. Vertex \((2,-4) ;\) opens upward C. Vertex \((4,-2) ;\) opens downward D. Vertex \((4,-2) ;\) opens upward E. Vertex \((-2,4) ;\) opens left F. Vertex \((-2,4)\); opens right G. Vertex \((-4,2) ;\) opens left H. Vertex \((-4,2) ;\) opens right $$y=-(x-2)^{2}-4$$

Step-by-Step Solution

Answer

A: Vertex \((2,-4)\); opens downward

Step 1: Identify the Parabola Form

Recognize that the given equation \( y = -(x-2)^{2}-4 \) is in the vertex form \( y = a(x-h)^{2} + k \). This form shows that the parabola opens upwards if \( a > 0 \) and downwards if \( a < 0 \), with the vertex at \((h, k)\).

Step 2: Extract Vertex and Direction

From the equation \( y = -(x-2)^{2}-4 \), identify \( a = -1 \), \( h = 2 \), and \( k = -4 \). The vertex is therefore \((2, -4)\), and since \( a = -1 \) is negative, the parabola opens downward.

Step 3: Match with Description

Compare the extracted vertex \((2, -4)\) and its upward/downward opening (downward in this case) to the appropriate description. The matching description in Column II is the one that states: "Vertex \((2, -4)\); opens downward", which corresponds to option A.

Key Concepts

Vertex Form

The vertex form of a parabolic equation is key to understanding its characteristics without graphing. It is expressed as \( y = a(x-h)^2 + k \). This form reveals critical aspects of the parabola:

• \( a \) determines the direction the parabola opens. If \( a > 0 \), it opens upward, and if \( a < 0 \), it opens downward.
• \( (h, k) \) is the vertex of the parabola. The vertex is the peak or the lowest point, depending on the direction of opening.

By rewriting any quadratic equation in this form, you can easily extract both the vertex and understand the opening direction of the parabola. This allows for quick analysis and matching with theoretical descriptions.

Equation Matching

Matching equations to their descriptions involves identifying components like the vertex and direction of the parabola's opening. Each piece of information in the vertex form points to a specific description:

• The vertex \((h, k)\) is derived directly from the equation in vertex form.
• The coefficient \( a \) denotes whether the parabola opens upwards or downwards.

In our exercise, the given equation was \( y = -(x-2)^2 - 4 \). From this, we extract that \( h = 2 \), \( k = -4 \), and \( a = -1 \). Therefore, our matching criteria will lead us to the description "Vertex \((2, -4)\); opens downward", which is a quick process once the vertex form is properly understood.

Graphing Parabolas

Graphing parabolas can be greatly simplified by understanding the vertex form. Each aspect of the equation gives visual cues:

• \( (h, k) \) is plotted directly on the coordinate grid as the vertex.
• If \( a > 0 \), the arms of the parabola rise upward from the vertex. If \( a < 0 \), the arms fall downward.
• The value of \( a \) affects the width of the parabola; larger absolute values of \( a \) narrow the parabola, while smaller values widen it.

These observations allow you to sketch the shape and orientation of the parabola promptly, providing a visual understanding to complement the algebraic interpretation.

Vertex Identification

Identifying the vertex from an equation in vertex form is a straightforward process. The vertex \((h, k)\) is found at the transformation components of the equation \( y = a(x-h)^2 + k \). Specifically:

• \( h \) is found by observing the shift along the x-axis. If the equation is \( y = a(x-2)^2 + k \), then \( h = 2 \).
• \( k \) is simply the constant at the end, representing the vertical shift.

Understanding how to extract the vertex provides a clear image of the parabola's highest or lowest point, depending on \( a \). This core concept ensures you can determine essential features of any quadratic function presented in vertex form.