Problem 47 - College Algebra

Question

For the following exercises, find the equation of the parabola given information about its graph. Vertex is (2,2)\(;\) directrix is \(x=2-\sqrt{2},\) focus is \((2+\sqrt{2}, 2)\)

Step-by-Step Solution

Answer

The equation of the parabola is \((y - 2)^2 = 4\sqrt{2}(x - 2)\).

Step 1: Identify the vertex and standard form

Since the vertex of the parabola is given as \((2, 2)\), the equation of the parabola in vertex form is \((y-k)^2 = 4p(x-h)\), where \((h,k)\) is the vertex of the parabola.

Step 2: Find the value of p

The directrix is \(x = 2 - \sqrt{2}\) and the focus is \((2 + \sqrt{2}, 2)\). The vertex is exactly halfway between the focus and the directrix. Therefore, the distance from the vertex \((2, 2)\) to the directrix \(x = 2 - \sqrt{2}\) or to the focus \((2 + \sqrt{2}, 2)\) is \(\sqrt{2}\), which is equal to \(|p|\). Thus, \(p = \sqrt{2}\).

Step 3: Write the complete equation

Using \(p = \sqrt{2}\), and the vertex \((h, k) = (2, 2)\), substitute into the vertex form equation: \((y - 2)^2 = 4\sqrt{2}(x - 2)\).

Step 4: Express the equation in a standard form if needed

The equation \((y - 2)^2 = 4\sqrt{2}(x - 2)\) might be suitable for some problems, but it can also be expanded if needed to standard quadratic form. However, for vertex equations as described, this step is usually not necessary unless specific reformatting is required.

Key Concepts

Vertex of Parabola

When we talk about the vertex of a parabola, it is helpful to imagine this point as the "center" of the parabola's "U." The vertex is a significant point because it is either the highest or lowest point on the graph, depending on which way the parabola opens. For our specific parabola, the vertex is at the point (2, 2).

• The vertex gives the parabola a specific location on the coordinate plane.
• This is the point \( (h, k) \) in the vertex form equation of a parabola.

By knowing the vertex, we not only get its position but also find a cornerstone for constructing the rest of the parabola's equation. The equation in vertex form can draw or predict how the parabola behaves. In mathematical terms, you often start with the vertex when you begin forming the equation of the parabola.

Directrix

The concept of the directrix is a bit intriguing but pivotal in defining a parabola. Imagine the directrix as an invisible line that the parabola "wants" to keep equidistant from its opposite side. For our parabola, the directrix is given by the line equation \( x = 2 - \sqrt{2} \).

• The directrix is always perpendicular to the axis of symmetry of the parabola.
• It works along with the focus to help define the shape and direction of the parabola.

The importance of the directrix comes into play when calculating the distance \( p \), which indicates how far the directrix is from the vertex. This is crucial because this same distance \( p \) will determine the shape and width of the parabola. Keeping these concepts in mind helps illustrate why the directrix is used in conjunction with the focus when we want to detail out a parabola fully.

Focus of Parabola

The focus is another important feature closely tied with the vertex and directrix. Picture the focus as a point that the parabola is "chasing." For our parabola, the focus is located at \( (2 + \sqrt{2}, 2) \).

• The focus, like the vertex, helps anchor the parabola along its path on a graph.
• The axis of symmetry passes through both the vertex and the focus.

One fascinating aspect of the focus is how it interacts with the rest of the parabola. Specifically, every point on the parabola is equidistant to both the focus and the directrix. This characteristic is an integral part of defining what makes a shape a parabola. It also explains why any attempt to find an equation of a parabola usually starts from understanding where the vertex and focus are.

Vertex Form Equation

The vertex form of a parabola equation is a highly efficient way to express a parabola and predicts its behavior. The vertex form is \( (y-k)^2 = 4p(x-h) \), where \( (h, k) \) is the vertex. For our current parabola example, the vertex is \( (2, 2) \) and \( p = \sqrt{2} \).

• Using these, the vertex form equation becomes \( (y-2)^2 = 4\sqrt{2}(x-2) \).
• "\((h, k)\)" pinpoints the vertex precisely, while \(p\) delineates the parabola's openness and direction.

One great advantage of the vertex form is its ability to quickly provide information on a graph's vertex position and direction. While in some scenarios the standard quadratic form might be more suitable, the vertex form is particularly insightful for understanding and visualizing the parabola's geometrical features.