Problem 42 - Precalculus with Calculus Previews

Question

Find an equation of parabola that satisfies the given conditions. Vertex \((-1,4),\) directrix \(x=0\)

Step-by-Step Solution

Answer

The equation of the parabola is \((y - 4)^2 = -4(x + 1)\).

Step 1: Understanding the Vertex Form

The vertex form of a parabola is given by the equation \( (y - k)^2 = 4p(x - h) \), where \((h,k)\) is the vertex of the parabola, and \(p\) is the distance from the vertex to the directrix, acting as the focal length. In this problem, the vertex is \((-1, 4)\).

Step 2: Determine the Direction and Value of 'p'

Since the directrix is vertical (\(x = 0\)), the parabola opens either to the left or right. Given that the vertex \((-1,4)\) is on the left of the directrix (since \(-1 < 0\)), the parabola opens towards the left. Thus, \(p\) should be negative.\(p\) is also the horizontal distance between the vertex and the directrix which is \(-1 - 0 = -1\).

Step 3: Substitute into the Vertex Form

Substituting \(h = -1\), \(k = 4\), and \(p = -1\) into the vertex form of the equation, we have: \((y - 4)^2 = 4(-1)(x + 1)\).

Step 4: Simplify the Equation

Simplify the equation by multiplying everything out: \((y - 4)^2 = -4(x + 1)\). This gives the equation of the parabola.

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is a popular method for writing the equation of a parabola. It focuses on the vertex, which is the most important point of a parabola, as it gives useful information about its shape and position. The vertex form equation is given as \[(y - k)^2 = 4p(x - h)\] where

• \((h, k)\) is the vertex of the parabola.
• \(p\) represents the focal length, or the distance from the vertex to the directrix.

This form is especially handy when the parabola opens either left or right. The vertex provides a point of symmetry for the parabola. By knowing the vertex and the focal length, you can easily sketch the parabola or transform it into other forms like the standard or general form.

Directrix of a Parabola

The directrix of a parabola is a fixed line used as a reference to measure the distance of each point on the parabola. Combined with the focus, the directrix helps us to define a parabola as the set of all points equidistant from both the focus and the directrix. The equation of the directrix can be either a vertical or a horizontal line, depending on the direction in which the parabola opens.In this particular exercise, the directrix is given as a vertical line \(x = 0\).

• If the directrix is vertical, the parabola opens left or right.
• If the directrix is horizontal, the parabola would open upwards or downwards.

Understanding the position of the directrix relative to the vertex is crucial as it determines the direction the parabola opens and affects the sign and value of \(p\), the focal length.

Focal Length of a Parabola

The focal length \(p\) of a parabola is a measure of how "wide" or "narrow" the parabola is. It represents the distance from the vertex to both the focus of the parabola and the directrix. In the vertex form equation \[(y - k)^2 = 4p(x - h)\],

• \(4p\) determines the "width" of the parabola.
• \(p\)'s sign tells us the direction of the parabola's opening.

For instance, in this exercise, \(p\) was calculated as \(-1\), indicating the parabola opens to the left. A larger absolute value of \(p\) indicates a wider parabola, while a smaller absolute value denotes a narrower one. Understanding \(p\) helps you grasp not just the direction of the parabola, but also its stretching.