Problem 14 - Precalculus: Mathematics for Calculus

Question

An equation of a parabola is given. (a) Find the vertex, focus, and directrix of the parabola. (b) Sketch a graph showing the parabola and its directrix. $$(y+1)^{2}=16(x-3)$$

Step-by-Step Solution

Answer

Vertex: (3, -1); Focus: (7, -1); Directrix: x = -1.

Step 1: Identify the Standard Form

The given equation is \((y + 1)^2 = 16(x - 3)\). This equation resembles the standard form of a parabola that opens horizontally, \((y - k)^2 = 4p(x - h)\). Here, we can identify \(k = -1\), \(h = 3\), and \(4p = 16\).

Step 2: Find the Vertex of the Parabola

The vertex \((h, k)\) of the parabola is \((3, -1)\). This is directly obtained from the rearranged vertex form of the equation.

Step 3: Calculate the Value of p

From the equation \(4p = 16\), solve for \(p\) to find \(p = \frac{16}{4} = 4\). Here \(p\) represents the distance from the vertex to the focus and from the vertex to the directrix.

Step 4: Determine the Focus

Since the parabola opens to the right, the focus \((h + p, k)\) is located at \( (3 + 4, -1)\). Thus, the focus is at \((7, -1)\).

Step 5: Identify the Directrix

The directrix of the parabola is a vertical line given by \(x = h - p\). Therefore, the directrix is the line \(x = 3 - 4 = -1\).

Step 6: Sketch the Graph

Sketch a coordinate plane. Plot the vertex at \((3, -1)\), the focus at \((7, -1)\), and the directrix as the vertical line \(x = -1\). Draw the parabola opening to the right, passing through the vertex and curving towards the focus, while maintaining symmetry relative to the vertex.

Key Concepts

Vertex

The vertex of a parabola serves as a critical point that showcases the parabola's most extreme value, whether minimum or maximum, depending on the parabola's orientation. In our specific case, the parabola is described by the equation \[(y + 1)^2 = 16(x - 3)\]This formula aligns with the horizontal form \[(y - k)^2 = 4p(x - h)\]In such a structure, the vertex is represented by the coordinates \((h, k)\).For the parabola in question:

• Here, both \(h\) and \(k\) are identified directly from the equation as \(h = 3\) and \(k = -1\).
• Thus, the vertex is located at \((3, -1)\).

The vertex is the midpoint between the focus and the directrix, playing a pivotal role in defining the parabola's shape and direction. In the sketch of this parabola, the vertex acts as the center point from which the curve bends, opening rightward due to the horizontal orientation. Recognizing this will aid in understanding the parabola's general look and behavior on the graph.

Focus

The focus of a parabola is a specific point that helps dictate the curve's shape. Parabolas are fascinating because they have reflective properties due to this focus, where rays parallel to the axis of symmetry converge. The distance from the vertex to the focus, denoted as \(p\), is vital and is calculated from the equation's standard form as \[4p = 16\] thus yielding \(p = 4\).To find the focus of our parabola:

• Given our vertex is \((3, -1)\) and since the parabola opens horizontally to the right, the focus is found using \((h + p, k)\).
• Replacing \(h = 3\), \(p = 4\), and \(k = -1\), the focus turns out to be at \((7, -1)\).

The focus lies inside the arc of the parabola, indicating the direction it opens is towards the focus. Understanding the focus helps in visualizing the parabolic path, as every point on the parabola is equidistant to this focus and a line known as the directrix.

Directrix

The directrix of a parabola is a line, providing a baseline opposite the focus. It is important in defining the parabolic curve, as it helps to balance the parabola with the focus. While the vertex is equidistant to both the focus and directrix, the formula for the directrix is \[x = h - p\]for horizontally opening parabolas.Considering our example:

• We have \(h = 3\) and \(p = 4\), so the directrix is the vertical line expressed as \(x = 3 - 4 = -1\).

The existence of the directrix as a vertical line helps maintain symmetry in the parabolic structure by dictating that each point on the parabola maintains an equal distance from both this line and the focus. This unique property is what assists a parabola in maintaining its graceful arc, stretching rightward. Understanding where the directrix lays provides insight into the size and spread of a parabola, as it essentially forms an invisible boundary that determines the path of the curve.