Problem 50 - College Algebra

Question

For the following exercises, find the equation of the parabola given information about its graph. Vertex is (1,2)\(;\) directrix is \(y=\frac{11}{3},\) focus is \(\left(1, \frac{1}{3}\right)\).

Step-by-Step Solution

Answer

The parabola's equation is \((x-1)^2 = -\frac{20}{3}(y-2)\).

Step 1: Understanding The Parabola's Structure

A parabola can be described with a vertex, a focus, and a directrix. The general vertex form of a parabola's equation is \( (x-h)^2 = 4p(y-k) \), where \((h,k)\) is the vertex and \(p\) is the distance from the vertex to the focus or directrix.

Step 2: Determine Parabola's Direction

Since the directrix and the focus have the same x-coordinate \(x=1\), the parabola opens vertically. The vertex \((1,2)\) is closer to the directrix \(y=\frac{11}{3}\) than to the focus \((1, \frac{1}{3})\), indicating that it opens downward.

Step 3: Calculate Distance \(p\)

The distance \(p\) is the distance from the vertex to the focus. Calculate \(p\) from the y-coordinates: \(p = 2 - \frac{1}{3} = \frac{6}{3} - \frac{1}{3} = \frac{5}{3}\).

Step 4: Identify the Sign of \(p\)

Since the parabola opens downward, \(p\) is negative. Thus, \(p = -\frac{5}{3}\).

Step 5: Write the Parabola Equation

Substitute \(h = 1\), \(k = 2\), and \(p = -\frac{5}{3}\) into the vertex form equation: \[(x-1)^2 = 4(-\frac{5}{3})(y-2)\]. Simplify to get, \[(x-1)^2 = -\frac{20}{3}(y-2)\].

Key Concepts

Vertex Form

The vertex form of a parabola is a useful way to express the equation of a parabola when you know its vertex. The vertex is the point where the parabola changes direction and is located at

• \((h, k)\)

, where \(h\) is the horizontal position and \(k\) is the vertical position. This form is

• \((x-h)^2 = 4p(y-k)\)

for vertical parabolas or \((y-k)^2 = 4p(x-h)\) for horizontal ones.
The constant \(p\) represents the distance from the vertex to the focus or the directrix, which is crucial in shaping the curve's openness. Remember,

• if \(p\) is positive, the parabola opens upwards or to the right
• if negative, it opens downwards or to the left.

Directrix

The directrix is a fixed line used in the geometric definition of a parabola. It is located opposite to the direction the parabola opens. For vertical parabolas, this line has a constant

• \(y-value\)

, and for horizontal parabolas, it has a constant

• \(x-value\)

. The formula for the directrix can be derived from the vertex form equation. The distance from the vertex to the directrix is the same as the distance from the vertex to the focus, represented by

• \(|p|\)

. In this exercise, since the given directrix is \(y = \frac{11}{3}\), it's a horizontal line indicating the parabola opens vertically.

Focus

The focus of a parabola is a single point such that the distance to any point on the parabola from this focus, is equal to the distance from the same point to the directrix. The focus always lies inside the parabola, meaning the parabola curves around it.
For vertical parabolas, if the vertex is

• \((h,k)\)

, and the parabola opens upwards or downwards, the focus will have the coordinates

• \((h, k + p)\)

or

• \((h, k - p)\)

. In this exercise, the focus is at

• \(\left(1, \frac{1}{3}\right)\)

, confirming the downward opening of the parabola when compared with the vertex at (1,2).

Equation of a Parabola

The equation of a parabola ties together its features: the vertex, the focus, and the directrix. In the vertex form,

• \((x - h)^2 = 4p(y - k)\)

is convenient for vertical parabolas and

• \((y - k)^2 = 4p(x - h)\)

for horizontal parabolas. Solving this involves knowing the vertex and calculating \(p\), the distance between the vertex and focus or directrix.
Substitute \(h\), \(k\), and \(p\) into the vertex form to get the explicit equation of the parabola. In our case:

• Given: vertex at \((1, 2)\), \(p = -\frac{5}{3}\)
• Equation: \[(x-1)^2 = -\frac{20}{3}(y-2)\]

. Keep in mind, a negative \(p\) indicates the parabola opens downwards or to the left, revealing the shape and direction of the parabola.