Problem 58 - 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. $$(x-2)^{2}=y+4$$ 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

B: Vertex \((2,-4)\); opens upward.

Step 1: Identify the Parabola Form

The given equation is \((x-2)^2 = y+4\). This is in the form \((x-h)^2 = y-k\), which represents a parabola that opens either upward or downward.

Step 2: Determine the Vertex

From the equation \((x-2)^2 = y+4\), we compare with the vertex form \((x-h)^2 = y-k\), and identify the vertex as \((h, k)\). Here \(h = 2\) and \(k = -4\), so the vertex is at \((2, -4)\).

Step 3: Determine Direction of Opening

Since the equation \((x-2)^2 = y+4\) is in the form \((x-h)^2 = y-k\), the parabola opens in the direction of the variable on the other side of the equation (the \(y\)-axis). This standard form \(x^2 = y\) represents a parabola opening upwards.

Step 4: Match with Description

The vertex is \((2, -4)\), and the parabola opens upwards according to our analysis. Thus, the correct description is B: Vertex \((2,-4)\); opens upward.

Key Concepts

Understanding Vertex Form

In the realm of quadratic functions, one of the most insightful forms to express a parabola is the vertex form. The vertex form of a parabola is typically written as \((x-h)^2 = 4p(y-k)\) when it involves the variable \(x\) squared. Here, \((h, k)\) is the vertex of the parabola, which acts as a critical point - the tip of the curve. This vertex provides a snapshot of the parabola's position on the graph.

Let's think of \((h, k)\) as coordinates on a map that direct us to where the parabola changes direction. In our example equation, \((x-2)^2 = y+4\), this translates directly to \((x-h)^2 = y-k\) when rearranged, showing \(h = 2\) and \(k = -4\). Therefore, the vertex is at \((2, -4)\). Changing these values in the expression affects where the vertex sits on the graph.

• \(h\) shifts the parabola along the x-axis.
• \(k\) shifts it along the y-axis.

In summary, the vertex form not only gives us the exact location of the vertex but also simplifies the process of graphing parabolas, making them much easier to understand and manipulate.

Exploring the Direction of Opening

The direction in which a parabola opens is crucial for understanding its behavior on a graph. Essentially, this is determined by the general orientation of the parabola. In the vertex form \((x-h)^2 = y-k\), the squared variable tells you the general direction of opening.

With \(x\) being squared, as in our equation \((x-2)^2 = y+4\), the parabola opens vertically. The direction—upward or downward—is determined by the sign and coefficient in front of the \(y-k\) portion.

• If it is positive, the parabola opens upwards.
• If it were negative, it would open downwards.

Since there's no negative sign affixed in our example, we deduce the parabola opens upwards. The importance of this lies in predicting the shape of the graph and understanding the nature of the solutions the quadratic equation provides.

An Overview of Quadratic Equations

Quadratic equations are fundamental in mathematics. They take the general form \(ax^2 + bx + c = 0\) and describe parabolas when graphed. The solutions to these equations are the values of \(x\) that make the equation true, and they can be found using a variety of methods such as factoring, completing the square, or the quadratic formula.

However, when dealing with equations like \((x-h)^2 = y-k\), we are examining a different expression of quadratic equations that demonstrate the shape rather than the roots. Here are some key insights:

• Quadratic equations can have no real roots, one root, or two distinct roots based on the discriminant.
• The graph of a quadratic function is called a parabola.
• In the context of vertex form, identifying the vertex and the direction of opening allows us to sketch the parabola accurately without needing to solve the equation.

These equations are essential not only in algebra but also in physics, engineering, and various fields where modeling and prediction of parabolic paths are necessary. Understanding quadratic equations in their many forms broadens one's capacity to tackle complex problems.