Problem 62 - 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 $$(y-2)^{2}=x+4$$

Step-by-Step Solution

Answer

Option H: Vertex (-4,2); opens right.

Step 1: Identify the Standard Form

Recognize that the equation \((y-2)^2 = x + 4\) is in the form \((y-k)^2 = 4p(x-h)\), which describes a horizontal parabola. This indicates that the parabola opens either to the left or to the right.

Step 2: Determine the Vertex

From the equation \((y-2)^2 = x + 4\), compare it with the standard form \((y-k)^2 = 4p(x-h)\). The vertex \((h, k)\) can be directly read as \((-4, 2)\), because the equation can be rewritten as \((y-2)^2 = x - (-4)\).

Step 3: Determine the Direction

Since the squared term is \((y-2)^2\) and it equals \(x + 4\), the parabola opens horizontally. Since \(x\) increases with \((y-2)^2\), it opens to the right (in a positive direction along the x-axis).

Step 4: Match with Description

The parabola described by the equation \((y-2)^2 = x + 4\) has a vertex at \((-4, 2)\) and opens to the right. Matching this with the options given, it corresponds to option H: Vertex \((-4,2)\) ; opens right.

Key Concepts

Vertex Form

In the world of parabolas, equations can appear in different forms. One of the most useful is the vertex form. The vertex form of a parabola that opens horizontally is written as \[(y - k)^2 = 4p(x - h)\]. Here,

• \( (h, k) \) is the vertex of the parabola, which is its highest or lowest point if it opens vertically and the rightmost or leftmost point if it opens horizontally.
• \( p \) is a constant that affects how wide or narrow the parabola is, and importantly, it tells us the direction in which the parabola opens.

It's crucial to identify the vertex from the equation directly by comparing it to the standard form. In our example equation, \((y-2)^2 = x + 4\), comparing with the vertex form reveals \(h = -4\) and \(k = 2\). Thus, the vertex is at \((-4, 2)\). Understanding the vertex forms equips you with a toolbox to identify key parabola features easily.

Parabola Direction

The direction in which a parabola opens is a vital piece of information. It helps us understand the shape and behavior of the graph. When dealing with a parabola in the horizontal vertex form \((y - k)^2 = 4p(x - h)\), the direction is determined by the sign and the positioning of \(p\). You can think of \(p\) as a pointer:

• If \(p\) is positive, the parabola opens to the right.
• If \(p\) is negative, the parabola opens to the left.

In our specific equation \((y-2)^2 = x + 4\), the coefficient in front of the \(x\) is effectively positive, indicating that our parabola opens to the right. Remembering these simple rules can guide you in determining the opening of any parabola just from its equation.

Equation Analysis

Analyzing an equation to understand a parabola involves dissecting its components. The equation \((y-2)^2 = x + 4\) is a clear example of a horizontally opening parabola. Let's break it down:

• Compare to standard form: By examining the equation, we identify it best fits the horizontal parabola form.
• Identify the vertex: We've determined the vertex to be at \((-4, 2)\) by comparing to the pattern \((y-k)^2 = 4p(x-h)\).
• Direction of opening: In the equation, \(x\) is isolated on one side, suggesting that as \((y-2)^2\) increases, \(x\) increases, indicating a rightward opening.

Analyzing equations systematically ensures you can match descriptions to equations accurately. Practice with different forms will make you proficient in quickly recognizing any parabola's properties from its equation.