Problem 32 - Precalculus: Functions and Graphs

Question

Find an equation of the parabola that satisfies the given conditions. $$\text { Vertex } V(4,7), \quad \text { focus } F(4,2)$$

Step-by-Step Solution

Answer

The equation of the parabola is \( y = -\frac{1}{20}(x-4)^2 + 7 \).

Step 1: Understand the Parabola Orientation

Since the vertex and focus have the same x-coordinate, the parabola opens vertically. Specifically, it must open downwards because the focus y-coordinate is less than the vertex y-coordinate.

Step 2: Use the Vertex Form of a Parabola

The vertex form for a vertically oriented parabola is \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex. Here \(h = 4\) and \(k = 7\), so initially we have:\[y = a(x-4)^2 + 7\]

Step 3: Calculate the Value of 'a' Using the Focus Position

The distance \( p \) from the vertex to the focus is needed, where \( p = \frac{1}{4a} \). The given distance is \( |7 - 2| = 5 \). Therefore, \( a \) can be found from:\[|p| = 5 = \frac{1}{4a} \]This results in:\[a = -\frac{1}{20}\]

Step 4: Write the Equation of the Parabola

Substitute \( a = -\frac{1}{20} \) into the vertex form equation, so the equation of the parabola is:\[y = -\frac{1}{20}(x-4)^2 + 7\]

Key Concepts

Vertex Form

The vertex form of a parabola is a useful way to express the equation of a parabola, especially when you know the vertex. The general expression for a vertically oriented parabola in vertex form is \( y = a(x-h)^2 + k \). Here:

• \((h, k)\) represents the vertex of the parabola.
• "a" is a coefficient that determines the direction and width of the parabola.

Knowing the vertex allows you to plug \( h \) and \( k \) directly into the formula to note how the parabola is anchored at the vertex point. For example, if your vertex is at (4, 7), your initial equation would look like \( y = a(x-4)^2 + 7 \). This doesn't give you the complete equation just yet but is a crucial first step in determining the parabola's shape and characteristics.

Focus of a Parabola

The focus of a parabola is a point that, along with the directrix, defines the set of points that make up the parabola. Each point on the parabola is equidistant from the focus and the directrix. For a parabola opening vertically, if the vertex is located at \((h, k)\) and the parabola focuses upwards, the focus will be \((h, k+p)\). If it focuses downwards, as in our example, the focus is below the vertex at \((h, k-p)\).
In the example provided, the vertex is at (4, 7) and the focus is at (4, 2). This tells us that the parabola opens downwards, as the y-coordinate of the focus is less than that of the vertex, which has significant implications for determining the parabola's orientation and equation.

Orientation of a Parabola

The orientation of a parabola refers to the direction in which it "opens". This direction is dictated by the relative positions of the vertex and the focus. In our example, both the vertex and focus align vertically (share the same x-coordinate), thus the parabola is vertically oriented. Whether it opens upwards or downwards depends on the relative positions of the vertex and focus:

• If the focus is above the vertex, the parabola opens upwards.
• If the focus is below the vertex, as in our case (vertex (4, 7) and focus (4, 2)), it opens downwards.

Understanding the orientation is crucial because it affects the sign of \( a \) in the vertex form equation, where a negative \( a \) indicates the parabola opens downwards. Consequently, this ensures that we calculate and apply the correct value of \( a \).

Distance from Vertex to Focus

The distance from the vertex to the focus, often symbolized as \( p \), is essential in constructing the equation of the parabola. It is used to determine the value of "a" in the vertex form equation via the formula \( a = \frac{1}{4p} \). In the given problem, the distance between the vertex (4, 7) and focus (4, 2) is 5 units. This distance helps calculate \( a \), knowing that the equation of a parabola can be described in terms of this distance.Given \( p = 5 \), we calculate \( a \) as follows:

• Since \( 5 = \frac{1}{4a} \), solve for \( a \).
• This results in \( a = -\frac{1}{20} \) for a downward opening parabola.

In practical terms, this means we've defined the "stretch" of the parabola. A smaller value of "a" indicates a wider parabola, while a larger value (in magnitude) points to a narrower shape. As the parabola opens downwards, "a" is negative.