Problem 59 - 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. $$y+2=-(x-4)^{2}$$ 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

C: Vertex (4,-2); opens downward.

Step 1: Identify the Form of the Parabola

The equation provided is \( y+2=-(x-4)^{2} \). This equation is in the form \( y - k = a(x - h)^{2} \), which represents a vertical parabola. We know it's vertical because \( y \) is expressed in terms of \( x^2 \).

Step 2: Determine the Vertex

In the general form \( y - k = a(x - h)^{2} \), the vertex is the point \((h, k)\). Comparing the given equation \( y+2 = -(x-4)^{2} \) to the standard form, we identify that \( h = 4 \) and \( k = -2 \), so the vertex is \((4, -2)\).

Step 3: Determine the Direction the Parabola Opens

The coefficient \( a \) in \( y - k = a(x - h)^{2} \) tells us the direction of the opening. Here \( a = -1 \), which is negative, so the parabola opens downward.

Step 4: Match with the Description

The correct match is the description with a vertex of \((4, -2)\) that opens downward. Referring to the options, the description that matches is option C: Vertex \((4,-2)\); opens downward.

Key Concepts

Vertex Form

The vertex form of a parabola's equation is incredibly useful for understanding the graphical representation of a quadratic function. When an equation is in the vertex form, it is structured as \( y - k = a(x - h)^2 \). Here, \( (h, k) \) denotes the vertex of the parabola, while \( a \) gives us clues about the parabola's orientation. Understanding the vertex form helps simplify the process of graphing parabolas without needing to convert the equation to the standard form.This form makes it easy to identify the vertex, which is a key point of the parabola. It indicates the peak or the lowest point, depending on whether the parabola opens upwards or downwards. Knowing the vertex form also allows for easier transformations, such as shifting the parabola along the graph or altering its shape by changing the value of \( a \). Using the vertex form as a tool helps students effectively graph and interpret parabolas with more confidence.

Vertex Identification

Identifying the vertex of a parabola is straightforward when using the vertex form \( y - k = a(x - h)^2 \). The vertex represents the point \( (h, k) \) in the equation, which is derived by comparing the given equation to the vertex format. For example, in the equation \( y+2=-(x-4)^2 \), this converts to \( y - (-2) = -(x - 4)^2 \), indicating a vertex at \( (4, -2) \). Understanding how to pinpoint this vertex is crucial as it guides you in sketching the parabola. The vertex shows where the highest or lowest part of the parabola is located on the graph. This technique aids in quick identification, allowing students to match equations with their descriptions without confusion. Through practice, recognizing the vertex becomes an intuitive step when working with parabolic equations.

Graphing Parabolas

Graphing parabolas can seem daunting at first, but it becomes manageable when using the vertex form of the equation. To graph a parabola, you start by identifying its vertex. From the vertex \( (h, k) \), you can then decide the width and direction of the parabola by looking at the value of \( a \).A few steps to keep in mind:

• Identify the vertex using the formula \( (h, k) \).
• Determine the sign of \( a \) to establish the direction—upward or downward.
• Consider the magnitude of \( a \). A larger value makes the parabola "steeper," while a smaller value produces a wider curve.
• Plot the vertex and sketch the parabola using the direction and width information.

By following these steps, students can accurately graph any parabola, thereby visualizing how changes in numbers affect the overall shape and position of the graph.

Parabola Direction

Understanding the direction in which a parabola opens is a fundamental part of graphing. The direction is dictated by the sign and the coefficient \( a \) in the vertex form \( y - k = a(x - h)^2 \). If \( a \) is positive, your parabola will open upwards, forming a "U" shape. Conversely, if \( a \) is negative, it will open downwards, like an inverted "U". Let's consider the equation \( y+2=-(x-4)^2 \). Here, \( a = -1 \), indicating that the parabola opens downward. This knowledge helps enormously when graph-matching exercises need to be completed quickly, as it gives you instant visual cues about the slope's general direction, either climbing out of the depth or descending into the trough.The direction also affects how you consider the maximum or minimum values of a quadratic equation. Upward-open parabolas have a minimum point at the vertex, while downward-open parabolas have a maximum point there.