Problem 27 - Precalculus with Limits

Question

In Exercises 19-32, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix: \(x = -1\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \(x = 4y^2\).

Step 1: Figure out the direction of the parabola

The directrix is given as \(x=-1\), which signifies the parabola opens to the right

Step 2: Determine the vertex form for the parabola

The vertex form for the parabola that opens to the right is \(x=4ap(y-h)^2+k\), but since the vertex is at origin, \(h\) and \(k\) are both equal to 0

Step 3: Determine the value of a

The absolute distance from the vertex to the directrix equals \(a\). Thus \(a = 1\)

Step 4: Writing the final equation

Substitute 1 into the equation \(x=4ap(y-h)^2+k\) to get the standard form of the parabola, \(x=4(1)(y)^2=4y^2\).

Key Concepts

Directrix of a Parabola

In the quest to understand the standard form of the equation of a parabola, knowing the role of the directrix is fundamental.

The directrix is a straight line that, along with the focus, helps to define a parabola. A parabola is the set of all points that are equidistant from the focus, a point, and the directrix, a line. Imagine the directrix as a mirror held parallel to the axis of symmetry of the parabola. Every point on the parabola is as far away from the focus as it is from this mirror. This property remains constant regardless of the parabola's orientation.

In our exercise, with the directrix given as \(x = -1\), it indicates that the parabola opens to the right, since the directrix is vertical. This clues us into the fact that the parabola's axis of symmetry is horizontal. If, contrastingly, the directrix was a horizontal line, the parabola would either open upwards or downwards. Recognizing the orientation is crucial as it affects the algebraic form the equation of the parabola will take.

Vertex of a Parabola

The vertex is another central characteristic of a parabola—it's the point where the parabola turns; it is the maximum or minimum point on the graph, depending on the direction the parabola opens.

In the given exercise, the vertex is at the origin (0,0). The coordinates of the vertex \((h, k)\) are pivotal when writing the equation of a parabola in vertex form, especially when the axis of symmetry is vertical, which can be expressed as \(y = a(x-h)^2 + k\), or horizontal, which our exercise indicates, expressed as \(x = 4ap(y-h)^2 + k\). Since the vertex is at the origin, \(h\) and \(k\) are both zero, which simplifies our equation significantly. The vertex's position at the origin also means that the equation will not have \(h\) and \(k\) components, making it easier for students to work with and understand.

Focus of a Parabola

If the directrix is one foundational piece, the focus is the other. The focus is a fixed point inside the parabola that, along with the directrix, defines the parabola’s reflective property. Each point on a parabola is positioned at an identical distance from the focus as it is from the directrix.

Following the symmetry of a parabola, the focus is always 'a' units away from the vertex along the axis of symmetry of the parabola. To find the focus, you will always use the value of 'a', which we have deduced is 1 in our exercise. In our situation where the parabola opens to the right, and the vertex is at the origin, the focus is at (1,0). This point is the ‘heart’ of the parabola; lines drawn from the focus to any point on the parabola will be reflected symmetrically across the parabola's axis.

The exact position of the focus gives us the '4a' part of the equation \(x = 4ap(y-h)^2 + k\), reflecting the squared part of a parabolic curve. Understanding the interaction between the vertex, focus, and directrix not only aids in graphing parabolas but also in grasping their applications, such as in satellite dishes and headlights.