Problem 3 - Precalculus

Question

Graph the parabolas. In each case, specify the focus, the directrix, and the focal width. Also specify the vertex. $$y^{2}=-8 x$$

Step-by-Step Solution

Answer

The vertex is (0, 0), the focus is (-2, 0), the directrix is x = 2, and the focal width is 8.

Step 1: Identify the Standard Form

The given equation is \( y^2 = -8x \). This is a quadratic equation in the form \( y^2 = 4px \), where \( 4p = -8 \). It represents a horizontal parabola that opens to the left.

Step 2: Find the Vertex of the Parabola

In the equation \( y^2 = -8x \), since it matches \( (y - k)^2 = 4p(x - h) \) but centered at the origin, we have \( h = 0 \) and \( k = 0 \). Therefore, the vertex is at (0, 0).

Step 3: Calculate the Value of p

From the equation \( 4p = -8 \), solve for \( p \) by dividing both sides by 4: \( p = -2 \). The negative p indicates that the parabola opens to the left.

Step 4: Determine the Focus

Using \( p = -2 \) and the definition that the focus is at \((h + p, k)\) for a horizontal parabola, substitute \( h = 0 \) and \( k = 0 \). The focus is at \((-2, 0)\).

Step 5: Identify the Directrix

The directrix of a horizontal parabola is given by \( x = h - p \). Substituting \( h = 0 \) and \( p = -2 \), the directrix is \( x = 2 \).

Step 6: Calculate the Focal Width

Focal width of a parabola is the absolute value of \( 4p \). Since \( 4p = -8 \), the focal width is \(|-8| = 8\).

Key Concepts

Focus

In the world of parabolas, the focus is a crucial point that helps describe the curve's orientation and properties. Imagine a spotlight or a lighthouse; the focus acts as a point from which light or sound emanates. For parabolas, this concept translates into a geometrical place where all the parabolic lines reflect from.
In our example, from the equation \( y^2 = -8x \), we found that the focus lies at \((-2, 0)\). Knowing that the general formula for the focus of a horizontally opening parabola (centered at the origin) is \((h + p, k)\), we substituted \( h = 0 \), \( k = 0 \), and \( p = -2 \) to find it.
It is crucial as it shows the direction of the parabola opens towards. In our case, it's pointing to the left, given the negative value of \( p \). The focus not only establishes the orientation but also influences how narrow or wide the parabola appears.

Directrix

The directrix complements the focus. Just as the focus is a direct point inside the parabola, the directrix is a line outside of it. This line acts as a reference from which the parabola maintains a constant distance.
Mathematically, for our parabola described by \( y^2 = -8x \), the directrix is found at \( x = 2 \). We determined this by using the formula \( x = h - p \), substituting \( h = 0 \) and \( p = -2 \). This yields the line \( x = 2 \) running vertically.
Interestingly, each point on the parabola is equidistant from both the directrix and the focus. Knowing the directrix allows us to understand more deeply the reflective property of parabolas, which is crucial in real-world applications like satellite dishes and headlights.

Vertex

The vertex of a parabola can be likened to its "central" point or peak. It is the spot where the curve starts, turns, or opens, depending on its orientation.
In our particular case of the equation \( y^2 = -8x \), the vertex is conveniently located at the origin point or \((0, 0)\). We know this because the equation matches the standard form \((y - k)^2 = 4p(x - h)\), where both \( h \) and \( k \) equal zero.
This location simplifies the plotting and understanding of the parabola's characteristics. The vertex provides the starting point from where all distances to the focus and directrix are measured, making it a pivotal feature when graphing and analyzing parabola behaviors.

Focal Width

The focal width is another vital feature of a parabola, representing the breadth of the parabola at its most central point. It describes how "opened" or "closed" the curve seems.

• This concept is directly tied to the parameter \( 4p \) in the parabola's standard form.
• For \( y^2 = -8x \), the computation gives a focal width of 8 since \(|4p| = |-8| = 8\).
• The absolute value is significant because it shifts our understanding from the direction of the opening to just how wide the parabola stretches.

Understanding the focal width gives you an advantage in sketching and comprehending the parabola geometrically. It offers insight into the curve's visual impact, which relates back to its practical uses, especially in optics and other fields where parabolic shapes are essential.