Problem 64 - A Graphical Approach to Precalculus with Limits

Question

Each equation in Exercises defines a parabola. Without actually graphing. match each equation with the appropriate description. $$x=-(y-2)^{2}-4$$ 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

Step-by-Step Solution

Answer

The correct description is G: Vertex \((-4, 2)\); opens left.

Step 1: Identify the Standard Form of a Parabola

The given equation is \( x = -(y-2)^2 - 4 \). This is an equation of a parabola in the form \( x = a(y-k)^2 + h \), where \( (h, k) \) is the vertex of the parabola and the sign of \( a \) determines the direction in which the parabola opens.

Step 2: Find the Vertex

In the equation \( x = -(y-2)^2 - 4 \), compare it to the standard form \( x = a(y-k)^2 + h \). From this comparison, we identify \( h = -4 \) and \( k = 2 \). Thus, the vertex of the parabola is \( (-4, 2) \).

Step 3: Determine the Direction the Parabola Opens

The coefficient \( a = -1 \) in \( x = -(y-2)^2 - 4 \), indicates that the parabola opens in the direction opposite to the direction of the positive x-axis. Since this parabola is in the form \( x = a(y-k)^2 + h \) and \( a \) is negative, the parabola opens to the left.

Step 4: Match the Description

From the vertex \((-4, 2)\) and the opening direction (left), compare it with the given options. The description that matches this information is: "G. Vertex \((-4, 2)\); opens left."

Key Concepts

Parabola

A parabola is a unique and distinct type of curve you often encounter in various mathematical contexts. It is characterized by its U-shaped structure on a graph. Parabolas can be identified by specific algebraic equations and are one of the fundamental shapes in the study of precalculus.

• Parabolas appear in equations as a quadratic term, either involving \(x^2\) or \(y^2\).
• They are fully determined by elements such as their vertex and the direction in which they open.

When studying parabolas, we primarily examine their standard forms, vertices, and opening directions, which provide crucial information about their graphical representation.

Vertex

The vertex of a parabola is a significant point where the curve turns. It can be thought of as the peak or the trough of the parabola depending on its orientation. Understanding the vertex is crucial as it tells us the position of the parabola on the coordinate grid.

• The vertex is represented as \((h, k)\) in the standard form of a parabola.
• It serves as a point of symmetry for the parabola and is the "highest" or "lowest" point when the parabola opens upwards or downwards.
• For sideways-opening parabolas, the vertex is still the central hinge but not necessarily a peak or trough.

Finding the vertex for a given parabola equation allows us to identify its exact position on the graph, enabling us to visually interpret or graph the curve easily.

Standard form of a parabola

The standard form of a parabola is essential for determining its shape and orientation. It offers a simplified way to understand and graph parabolas effectively. The equations can vary depending on whether the parabola opens upwards, downwards, left, or right.

• The general standard form for a vertical parabola is \(y = ax^2 + bx + c\).
• For horizontal parabolas, it can be written as \(x = a(y-k)^2 + h\), where \( (h, k) \) is the vertex.
• The coefficient \(a\) in the equation affects the width and direction of the opening of the parabola.

Analyzing equations in their standard forms enables you to recognize key features like vertices and the axis of symmetry, as well as understand how changes in parameters affect the graph of the parabola.

Opening direction of a parabola

The opening direction of a parabola is dictated by the coefficient \(a\) in its standard form. It defines whether the parabola opens up, down, left, or right. Knowing which direction a parabola opens helps in predicting its behavior and sketching its graph accurately.

• For vertical parabolas (\(y = ax^2 + bx + c\)), if \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.
• In horizontal parabolas (\(x = a(y-k)^2 + h\)), a positive \(a\) means it opens to the right, while a negative \(a\) means it opens to the left.
• The value of \(a\) not only affects direction but also influences the "width" or "narrowness" of the parabola.

Understanding the direction in which a parabola opens is crucial for graph interpretation and solving mathematical problems accurately.