Problem 63 - 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 $$x+2=-(y-4)^{2}$$

Step-by-Step Solution

Answer

E. Vertex (-2,4); opens left

Step 1: Identify the Standard Form

The given equation is \(x + 2 = -(y - 4)^2\). This equation matches the standard form of a parabola opening sideways, \((y - k)^2 = 4p(x - h)\), where the vertex is \((h, k)\) and the direction the parabola opens is determined by the sign of \(4p\). Here, \(4p = -1\) indicating the parabola opens to the left.

Step 2: Determine the Vertex

From the equation \(x + 2 = -(y - 4)^2\), we rewrite it to identify the vertex: \(x - (-2) = -(y - 4)^2\). This shows that the vertex is \((-2, 4)\).

Step 3: Determine the Direction

Since the equation is \(x + 2 = -(y - 4)^2\) and has a negative sign in front of \((y - 4)^2\), the parabola opens to the left. In the equation \( (y - k)^2 = 4p(x - h)\), if \(p\) is negative, it opens to the left.

Key Concepts

Vertex Form

The vertex form of a parabolic equation is a very useful way to represent a parabola because it provides immediate insight into the position of its vertex, and thus its graph. A parabolic equation is in vertex form when it looks like this: \( y = a(x-h)^2 + k \). In this expression:

• \((h, k)\) is the vertex of the parabola.
• The value of \(a\) controls how "wide" or "narrow" the parabola appears, as well as its direction.

Knowing the vertex helps in quickly sketching the parabola and understanding its orientation. Consider the transformation from the origin position of a parabola to its vertex position when adjusting \((h, k)\). The \(h\) value moves the graph horizontally, while \(k\) moves it vertically. This simplicity makes the vertex form a popular choice for graphing or analyzing parabolas.

Direction of Parabolas

The direction in which a parabola opens is an important feature of its graph. This is determined by the sign of the coefficient that multiplies the square term in its equation.

• If the equation is in the form \((y-k)^2 = 4p(x-h)\), and \(p > 0\), the parabola opens to the right.
• If \(p < 0\), the parabola opens to the left.
• For equations like \(y = ax^2 + bx + c\), if \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards.

In our example equation \(x + 2 = -(y - 4)^2\), the negative sign before \((y - 4)^2\) tells us that the parabola opens to the left. This is because it matches the sideways opening parabola form, indicating a horizontal orientation. Understanding the direction helps predict and visualize the shape of the parabola even without graphing.

Standard Form of Parabolas

The standard form of a parabola is one of the simplest representations, often making it easier to understand its key elements. A standard form equation of a parabola might be \(y = ax^2 + bx + c\) for vertical parabolas or \((y-k)^2 = 4p(x-h)\) for horizontal ones. The latter form is useful to:

• Identify the vertex \((h, k)\),
• Determine the direction it opens.

In our problem's equation, \(x + 2 = -(y - 4)^2\), it follows the horizontal standard form. By rearranging, \((y - 4)^2 = -1(x + 2)\), we can deduce:

• The vertex is at \((-2, 4)\).
• The coefficient of \'\(x + 2\)' being negative \((-1)\) indicates an opening to the left.

This form is powerful in solving practical problems where understanding the parabolic shape's constitution and orientation matters for applications in fields such as physics, engineering, and statistics.