Problem 61 - 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 $$(y-4)^{2}=x+2$$

Step-by-Step Solution

Answer

The description is F: Vertex \((-2,4)\); opens right.

Step 1: Identify the Standard Form

The given equation is \((y-4)^2 = x + 2\). This matches the standard form of a parabola \((y-k)^2 = 4p(x-h)\), which indicates a horizontal parabola.

Step 2: Determine Vertex

In the standard form \((y-k)^2 = 4p(x-h)\), the vertex is \((h, k)\). Here, we compare \((y-4)^2 = x+2\) to find that \(h = -2\) and \(k = 4\), so the vertex is \((-2, 4)\).

Step 3: Determine Opening Direction

For \((y-k)^2 = 4p(x-h)\), if \(p > 0\), the parabola opens to the right. If \(p < 0\), it opens to the left. In the equation \((y-4)^2 = x+2\), the coefficient of \(x+2\) is positive, thus it opens to the right.

Step 4: Match the Description

From Step 2 and Step 3, we concluded the parabola has vertex \((-2, 4)\) and opens to the right. This matches description F: Vertex \((-2, 4)\) opens right.

Key Concepts

Vertex Form

Understanding the vertex form of a quadratic equation is crucial for identifying key characteristics of parabolas. The vertex form is typically expressed as \[(y - k)^2 = 4p(x - h)\] for horizontal parabolas, or \[(x - h)^2 = 4p(y - k)\] for vertical ones. In this format,

• \( (h, k) \) represents the vertex of the parabola.
• The parameter \( p \) is related to the distance between the vertex and the focus of the parabola, which also influences the opening direction.

For instance, given the equation \((y-4)^2 = x + 2\), we can directly identify the vertex as \((-2, 4)\) by comparing it to the form \((y-k)^2 = 4p(x-h)\). Recognizing the vertex provides a starting point for further analysis of the parabola's properties.
Knowing the vertex is vital because it indicates the peak or turning point of the parabola, offering insight into its shape and position on a graph.

Opening Direction

The opening direction of a parabola can be determined by examining the sign of the variable or coefficient next to the variable that isn't squared. This outline helps recognize whether a parabola opens upwards, downwards, left, or right.

For a horizontal parabola like \((y-k)^2 = 4p(x-h)\):

• If \( p > 0 \), it opens to the right.
• If \( p < 0 \), it opens to the left.

If you consider our example equation \((y-4)^2 = x + 2\), it opens to the right because the term on the right side of the equation \((x+2)\) is positive.
Understanding the opening direction is crucial, especially in calculations requiring the prediction of intersections or how the parabola will interact with other geometric figures.

Quadratic Equations

Quadratic equations form the backbone for understanding parabolas and their properties. They are expressed in various forms such as the vertex, standard, or factored forms, and they represent the equations of parabolas, identifiable by their unique U-shape in graphs.
In the context of quadratic equations, parabolas can be expressed in a way that distinctively showcases their properties, such as their vertex and opening. For instance, the general form \( ax^2 + bx + c = 0 \) can be converted into the vertex form by completing the square, allowing for easy visualization of critical attributes like the vertex and opening direction.

• Quadratic equations often lead to two solutions, depicting where the parabola intersects the x-axis.
• They can be used to model various practical scenarios, such as projectile motion, where the path followed is a parabola.

Grasping quadratic equations equips students not only with the knowledge of solving them algebraically but understanding the geometric interpretations as well.