Problem 86 - A Graphical Approach to Precalculus with Limits

Question

Find an equation of a parabola that satisfies the given conditions. Focus \((-1,2) ;\) vertex \((3,2)\)

Step-by-Step Solution

Answer

The equation of the parabola is \((y - 2)^2 = -16(x - 3)\).

Step 1: Understanding Parabola Orientation

First, identify the orientation of the parabola based on the given coordinates of the focus \((-1, 2)\) and the vertex \((3, 2)\). Since both share the same y-coordinate, the parabola opens horizontally, hence it opens either left or right.

Step 2: Calculating the Distance Between the Vertex and Focus

Identify the distance between the vertex and focus horizontally: \[ p = |3 - (-1)| = 4 \].

Step 3: Determining Direction of Opening

As the vertex \((3, 2)\) is to the right of the focus \((-1, 2)\), the parabola opens to the left. Thus, the value of \( p \) is negative, \( p = -4 \).

Step 4: Using the Standard Form of a Horizontal Parabola

For a horizontal parabola with vertex \((h, k)\), the equation is \( (y-k)^2 = 4p(x-h) \). Substitute \( h = 3 \), \( k = 2 \), and \( p = -4 \):\[ (y - 2)^2 = 4(-4)(x - 3) \]

Step 5: Simplifying the Equation

Simplify the equation from the previous step:\[ (y - 2)^2 = -16(x - 3) \]

Key Concepts

Focus and Vertex

Understanding the focus and vertex of a parabola is crucial when dealing with its equations. In this case, the focus is located at the coordinates \((-1, 2)\) and the vertex is at \((3, 2)\). The focus is a point that lies on the axis of symmetry of the parabola, and the vertex is the midpoint of the parabola’s curvature.

Here's how to interpret the information:

• The vertex represents the "turning point" of the parabola. It's where the parabola changes direction.
• The focus directs how "wide" or "narrow" the parabola opens, as it’s a point around which the parabola curves.

When connecting these two points, you can determine how the parabola is oriented and where it opens, leading us to its equation.

Parabola Orientation

Orientation tells us which direction the parabola opens. From the given coordinates of the focus and vertex, it's clear that the parabola described in this exercise opens horizontally. Both the focus and vertex share the same y-coordinate \((y = 2)\), indicating that the parabola is oriented along the horizontal axis rather than the vertical one.

To identify the direction in which the parabola opens:

• If the focus is to the left of the vertex, the parabola opens left.
• If the focus is to the right of the vertex, it opens right.

In this exercise, the focus \((-1, 2)\) lies to the left of the vertex \((3, 2)\), which means the parabola opens to the left.

Distance Calculation

Calculating the distance between the vertex and the focus helps us understand the parameter \( p \) in the parabola's equation. The distance, denoted as \( p \), determines how "steep" the parabola opens. It is simply the absolute distance between the vertex and the focus along the axis of symmetry.

In the exercise, calculate \( p \) as follows:

• For points on a horizontal line: \( p = |3 - (-1)| = 4 \).

After calculating the direction of opening, note that since the parabola opens to the left, \( p \) is negative. Therefore, \( p = -4 \).

The value of \( p \) is plugged into the parabola's equation to depict how far the vertex is from the center, guiding its shape and curvature.

Standard Form of Parabola

The equation representing a parabola depends on its orientation and vertex. For horizontal parabolas, the standard form is \((y-k)^2 = 4p(x-h)\), where \((h, k)\) is the vertex, and \(p\) is the distance derived from previously mentioned calculations.

Inserting the given values into the standard form involves these steps:

• Substitute \( h = 3 \), \( k = 2 \), and \( p = -4 \).
• This results in the equation: \((y - 2)^2 = -16(x - 3)\).

Simplifying this equation gives a clear representation of the parabola in standard form. Knowing the standard form allows easy graphing and understanding of the parabola's fundamental characteristics, such as orientation and direction of opening.