Problem 19 - College Algebra

Question

For the following exercises, rewrite the given equation in standard form, and then determine the vertex \((V),\) focus \((F),\) and directrix \((d)\) of the parabola. $$ (y-4)^{2}=2(x+3) $$

Step-by-Step Solution

Answer

Vertex: \((-3, 4)\), Focus: \((-\frac{5}{2}, 4)\), Directrix: \(x = -\frac{7}{2}\).

Step 1: Expand the Equation

The initial equation is given as \((y-4)^{2}=2(x+3)\). To rewrite it, let's start by expanding both sides. However, the equation is actually already simplified for converting, so next we will complete the square on the equation to make the expression more visible.

Step 2: Rewrite in Standard Form

The standard form for a parabola opening horizontally is \((y-k)^2 = 4p(x-h)\). For the equation \((y-4)^2 = 2(x+3)\), compare it with the standard form. We identify \(h = -3\), \(k = 4\), and \(4p = 2\), thus \(p = \frac{1}{2}\). The equation is already in standard form as \((y-4)^2 = 2(x+3)\).

Step 3: Determine the Vertex

The vertex form of the parabola apart from standard form gives the vertex directly. Here, it's \((h, k)\). From the equation \((y-4)^2 = 2(x+3)\), the vertex \(V\) is \((-3, 4)\).

Step 4: Calculate the Focus

To find the focus \(F\) of the parabola, use the relationship \((h+p, k)\). Here, \(h = -3\) and \(p = \frac{1}{2}\). Thus, the focus is located at \((-3 + \frac{1}{2}, 4) = (-\frac{5}{2}, 4)\).

Step 5: Determine the Directrix

The directrix of a parabola in a horizontal orientation is located at \(x = h - p\). So, with \(h = -3\) and \(p = \frac{1}{2}\), the directrix is \(x = -3 - \frac{1}{2} = -\frac{7}{2}\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point because it represents the tip of the curve, which is either the highest or lowest point, depending on its orientation. In simpler terms, it's like the 'center' of the parabola.
From our given equation \((y-4)^2 = 2(x+3)\), identified in the standard form for a parabola, the vertex is directly found as \((h, k)\). Here, \(h = -3\) and \(k = 4\), resulting in the vertex being at \((-3, 4)\).

• The vertex form directly gives the coordinates \((h, k)\).
• For parabolas that are not perfectly vertical or horizontal, converting to another form may be needed.

Focus of a Parabola

The focus of a parabola is a significant point because it plays an essential role in its geometric definition. It is used along with the directrix to define the set of points (locus) that form a parabola. All the points on the parabola are equidistant from the focus and the directrix.
In the equation provided \((y-4)^2 = 2(x+3)\), we can use the identified value of \(p\) which is \(\frac{1}{2}\). The relationship is \((h+p, k)\) for horizontal parabolas, so we substitute to find the focus:

• Using \(h=-3\) and \(p=\frac{1}{2}\), the focus = \(-3 + \frac{1}{2} = -\frac{5}{2}\).
• The y-coordinate remains the same, so the focus: \((-\frac{5}{2}, 4)\).

Directrix of a Parabola

The directrix of a parabola is a line that serves as a reference for the curve. It ensures the reflective property of the parabola along with the focus. Every point on the parabola is equidistant to the focus and the directrix.
For our horizontally oriented parabola given as \((y-4)^2 = 2(x+3)\), the directrix is located at \(x = h - p\). Let's calculate this for our equation:

• With \(h = -3\) and \(p = \frac{1}{2}\), the directrix: \(x = -3 - \frac{1}{2} = -\frac{7}{2}\).
• It will be a vertical line since \(x = \text{constant}\).

Standard Form of a Parabola

The standard form of a parabola's equation simplifies finding its vertex, focus, and directrix. It helps rewrite the parabola equation to easily identify these elements. Generally, for horizontal parabolas, the standard form is \((y-k)^2 = 4p(x-h)\).
In our problem, the given equation \((y-4)^2 = 2(x+3)\) is already in the standard form. Here's how we verified and arranged it:

• Comparing it with \((y-k)^2 = 4p(x-h)\), we have \(h = -3\), \(k = 4\), and \(4p = 2\).
• This leads to determining \(p = \frac{1}{2}\), important for finding focus and directrix.

Standard form equations offer a clear path to analyze and understand the parabola's characteristics effortlessly.