Problem 88 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the parabola with the given characteristics. Vertex: (-2,1)\(;\) directrix: \(x=1\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola with vertex (-2,1) and directrix \(x=1\) is \(y^2 - 2y + 12x + 25 = 0\).

Step 1: Determine the direction of the parabola.

Note where the directrix is. When the directrix is a vertical line (given by \(x = a\)), this indicates that the parabola will open towards the left or right, not upwards or downwards.

Step 2: Decide on the parabola's opening direction - left or right.

The parabola's vertex is the point (-2,1), which lies to the left of the directrix x = 1. The parabola opens in the direction opposite to that of its directrix, so in this case, the parabola opens to the left.

Step 3: Find the value of \(a\).

The value of \(a\) in the standard form of a parabola equation represents the distance from the vertex to the directrix or from the vertex to the focus. In this case, the vertex at (-2, 1) is 3 units away from the directrix at \(x = 1\). The negative sign indicates the parabola opens to the left.

Step 4: Formulate the equation of the parabola.

So, now we have all we need to formulate the equation of the parabola. The vertex coordinates represent \((h,k)\), and \(a\) is -3. The parabola opens horizontally so the equation is \((y-k)^2 = 4a(x-h)\). Substitute the known values.

Step 5: Finalize the equation

After substituting the values, the equation becomes \((y - 1)^2 = 4(-3)(x + 2)\). Simplifying this yields \((y-1)^2 = -12(x + 2)\). Now convert this into standard form by expanding and simplifying: \(y^2 - 2y + 1 = -12x - 24\), thus \(y^2 - 2y + 12x + 25 = 0\).

Key Concepts

Standard Form

The standard form of a parabola is a specific equation that helps us identify its shape and position easily. For a parabola opening horizontally, the standard form is

• \((y - k)^2 = 4a(x - h)\)

where

• \((h, k)\) represents the vertex of the parabola, indicating its highest or lowest point.
• \(a\) is a constant that controls the width and direction of the parabola's opening.

In our exercise, the vertex is located at \((-2,1)\) and the parabola opens to the left because the directrix is located at \(x = 1\). By using the distance between the vertex and the directrix and noting the direction, you can calculate the value of \(a\), which here is -3. Thus, the standard form of this particular parabola is derived as \[ (y-1)^2 = -12(x+2)\] indicating it opens to the left with a narrower spread due to the negative and larger magnitude of \(a\).
This form is especially useful in mathematical contexts where horizontal parabolas are analyzed, providing insight into the parabola’s properties and behaviors.

Directrix

The directrix is an essential component of understanding a parabola's geometry. It is a fixed line that helps determine the direction in which the parabola opens. In parabolas, the orientation of the directrix plays a crucial role:

• If the directrix is vertical (like "\(x = 1\)") the parabola will open horizontally.
• If the directrix is horizontal, the parabola will open vertically.

In this exercise, the directrix \(x = 1\) indicates that the parabola opens horizontally but toward the left, because the vertex \((-2,1)\) is situated to the left of \(x = 1\). The placement and relationship of the vertex and directrix allow us to harness symmetry, providing control over the calculus and geometry for plotting parabolas effectively. The perpendicular distance from vertex to the directrix also provides the magnitude for the term \(4a\) in the standard equation, especially impacting how narrow or wide the parabola spreads.

Vertex Form

The vertex form of a parabola is a convenient way to express its equation, especially when you already know the vertex. This form is directly derived from the concept of shifting a standard parabola, which simplifies visualizing or graphing it.
The vertex form for a vertical parabola is given as:

• \((x - h)^2 = 4a(y - k)\)

For a horizontal parabola, as in our case:

• \((y-k)^2 = 4a(x-h)\)

Here,

• \((h, k)\) represents the coordinates of the vertex.
• \(a\) is the parameter affecting the width and direction of the parabola.

In our exercise, leveraging the vertex form allows for a straightforward transition to finding the equation of the parabola: \[ (y-1)^2 = 4(-3)(x+2)\]This is a vital step when moving between different forms of a parabolic equation, ensuring accuracy and ease in representation within various contexts.

Focus and Directrix Relationship

The relationship between the focus and the directrix is fundamental in understanding parabolic shapes.
This geometry is defined such that:

• Every point on the parabola is equidistant to the focus and the directrix.

This creates the iconic "U-shape" of a parabola. In mathematical terms, if we have a vertex \((h, k)\), and the distance \(a\) from the vertex to the directrix or to the focus, then

• The focus will be located \(a\) units from the vertex on the parabolic curve.

For a horizontal parabola:

• If it opens right, the formula for focus with \((h - a, k)\)
• If it opens left, as in this exercise, it's described by \((h + a, k)\)

By understanding this delicate balance of distances, it becomes apparent how the vertex and directrix help determine overall parabolic behavior, aiding in precise graphing and applications across physics and engineering. Defining these attributes empowers one to effectively analyze real-world parabolic structures, such as satellite dishes and suspension bridges.