Problem 37 - Precalculus with Calculus Previews

Question

Find an equation of parabola that satisfies the given conditions. Focus \((8,-3),\) vertex (0,-3)

Step-by-Step Solution

Answer

The equation of the parabola is \( (y + 3)^2 = 32x \).

Step 1: Understand the Components of a Parabola

A parabola's standard form can be described as \( y = a(x - h)^2 + k \) where \((h,k)\) is the vertex of the parabola. Additionally, the focus of a parabola gives us information about the direction and width of the parabola.

Step 2: Given Conditions

We have been given the vertex of the parabola at \((0, -3)\) and the focus at \((8, -3)\). The focus and vertex share the same y-coordinate, indicating this is a horizontally oriented parabola.

Step 3: Determine Orientation and Direction

Since the vertex and focus have the same y-coordinate, this implies that the parabola opens horizontally. Given that the focus \((8, -3)\) is to the right of the vertex \((0, -3)\), the parabola opens to the right.

Step 4: Use the Vertex Form of a Parabola that Opens Horizontally

The equation for a horizontally opening parabola can be written as \( (y - k)^2 = 4p(x - h) \). Here \(h = 0\), \(k = -3\), and \((h + p, k) = \text{focus}\), so \(h + p = 8\), giving us \(p = 8\).

Step 5: Construct the Equation

Substituting the known values into the equation \((y + 3)^2 = 4p(x)\), we have \( (y + 3)^2 = 32x \) as the equation of the parabola.

Key Concepts

Vertex Form of Parabola

The vertex form of a parabola is a way to express the quadratic equation of a parabola to easily identify its key features. It is useful for quickly finding the vertex of a parabola and understanding how shifts and stretches affect its graph. The formula is expressed as \( y = a(x - h)^2 + k \), where \((h, k)\) represents the vertex of the parabola.

• \(h\) is the horizontal shift from the origin.
• \(k\) is the vertical shift from the origin.
• \(a\) determines the width and the direction of the opening (whether it opens upwards or downwards).

In the context of our exercise, the vertex form helps us start constructing the equation of a parabola when given certain conditions, such as the vertex point and orientation. Understanding this form is pivotal as it simplifies solving and graphing functions that follow this curve.

Focus and Directrix

A parabola is uniquely defined by its focus and directrix, fundamental features that give the parabola its shape. The focus is a fixed point that combines with the directrix, a fixed line, to generate the curve. Every point on a parabola is equidistant from both the focus and directrix.

• The focus guides the direction in which the parabola opens.
• The vertex, focus, and directrix are all interdependent in determining the parabola's exact shape.

In the exercise, knowing the focus at \((8, -3)\) and the vertex at \((0, -3)\) allowed us to glean that our parabola must open horizontally. With this information, we can derive and verify the equation itself. Focus and directrix play an essential role, especially when dealing with conic sections, as they dictate orientation and position.

Horizontal Parabola Orientation

A horizontal parabola is one that opens left or right rather than up or down. This orientation flips certain aspects of the standard vertical form of a parabola, making the axis of symmetry horizontal instead of vertical. The equation shifts slightly to accommodate the orientation change: \[ (y - k)^2 = 4p(x - h) \]

• \( (h, k) \) is the vertex of the parabola.
• \(4p\) relates to the parabola’s width and direction; \(p\) is the distance from the vertex to the focus.

In our example, with both the vertex and focus sharing a y-coordinate, we know the parabola opens horizontally. This exercise shows that understanding the orientation is crucial in determining the correct structure and form of the equation. Recognizing the direction in which the parabola opens allows us to utilize the appropriate formula and ensure the parabola behaves as defined by its geometric properties.