Problem 30 - College Algebra with Corequisite Support

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. \(x^{2}+4 x+8 y-4=0\)

Step-by-Step Solution

Answer

The vertex is \((-2, 1)\), focus is \((-2, -1)\), and directrix is \(y = 3\).

Step 1: Rearrange the Equation

Start by rearranging the terms of the given equation \(x^2 + 4x + 8y - 4 = 0\) to isolate the quadratic terms. The equation can be written as \(x^2 + 4x = -8y + 4\).

Step 2: Complete the Square

To complete the square for the \(x\) terms, take the coefficient of \(x\), which is 4, divide it by 2 to get 2, and then square it to get 4. Add and subtract 4 on the left side: \(x^2 + 4x + 4 - 4 = -8y + 4\), which simplifies to \((x + 2)^2 - 4 = -8y + 4\).

Step 3: Simplify and Rearrange

Move \(-4\) to the right side of the equation: \((x + 2)^2 = -8y + 8\). Simplify to get \((x+2)^2 = -8(y-1)\).

Step 4: Identify Vertex, Focus and Directrix

The equation is now in the form \((x-h)^2 = 4p(y-k)\), where \((h,k)\) is the vertex. From \((x+2)^2 = -8(y-1)\), we identify \(h = -2\) and \(k = 1\), so the vertex \(V\) is \((-2, 1)\). The value of \(4p = -8\) gives \(p = -2\). Thus, the focus \(F\) is \((-2, 1-2) = (-2, -1)\), and the directrix \(d\) is \(y = 1 + 2 = 3\).

Key Concepts

Vertex

The vertex of a parabola is a fundamental concept as it represents the point where the curve changes direction. It is the "peak" or "valley" of the parabola, depending on its orientation. For the equation of a parabola written in standard form \((x-h)^2 = 4p(y-k)\),\(\hspace{0.25 cm}(h, k)\) is the vertex of the parabola.

In our example equation, which eventually becomes \((x+2)^2 = -8(y-1)\), the vertex can be easily identified by comparing it to the standard form. Here, \(h = -2\) and \(k = 1\), which means the vertex \(V\) is \((-2, 1)\).

• Vertex as a Minimum or Maximum: When a parabola opens upwards, the vertex is at its minimum point. When the parabola opens downwards, the vertex is at its maximum point.
• Shifting the Vertex: The values of \(h\) and \(k\) in the equation indicate how the vertex is shifted from the origin. A positive \(h\) moves it right, and a positive \(k\) moves it up.

Focus

The focus of a parabola is a key point that, along with the directrix, helps in defining the parabola's shape. It lies on the axis of symmetry of the parabola and is located \(|p|\) units away from the vertex.

In our case, the equation \((x+2)^2 = -8(y-1)\) gives us \(4p = -8\), which implies \(p = -2\). This tells us that:

• Since \(p\) is negative, the parabola opens downwards.
• The focus is located at \((-2, 1 - 2)\) which simplifies to \((-2, -1)\).

The concept of a focus can be best understood through visualization. Imagine how every point on the parabola is equidistant to this focus point and the directrix. This property, called the reflective property of parabolas, is why parabolas can direct sound and light to the focus point.

Directrix

The directrix of a parabola is a straight line that is used to define and construct the parabola along with the focus. It is located opposite the focus, and the parabola is defined as all the points equidistant from the focus and the directrix.

From the equation \((x+2)^2 = -8(y-1)\), we have already determined that \(p = -2\). Therefore, the directrix is a line parallel to the horizontal axis of symmetry and is located at a distance \(|p|\) from the vertex. Since the parabola opens downwards, and vertex \(V\) is at \((-2, 1)\), the directrix is found by calculating \(y = 1 + 2\), which results in:

• The directrix is the line \(y = 3\).

The directrix plays an essential role in the geometry of a parabola. It ensures that any point \(P\) on the parabola keeps the equation of a parabola by maintaining its distance from the focus and the directrix equal. Consider how:

• The distance from any point on the parabola to the directrix is equal to the perpendicular distance from it to the focus.
• It acts as a balancing line ensuring symmetry in the parabola.