Problem 24 - 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. Focus: \((0, -2)\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \(y=-(1/8)x^2\)

Step 1: Determine orientation and a-value

The focus of the parabola is under the vertex (0, 0). When the focus is above or below the vertex, the parabola has a vertical orientation, meaning the general form will be \((x-h)^2 = 4a(y-k)\). Since the vertex and focus lie on the y-axis, the value of \(a\) will be the absolute y-coordinate of the vertex subtracted from the y-coordinate of the focus, i.e., \(-2-0 = -2\). So \(a=-2\)

Step 2: Substituting vertex and a into general form

Substitute the vertex (h, k = 0, 0) and \(a=-2\) into \((x-h)^2 = 4a(y-k)\) to get \(x^2 = -8y\)

Step 3: Final Equation of Parabola

Re-arranging the equation from step 2, we have \(y=-(1/8)x^2\). So the standard form of the equation of the parabola that opens downwards and has its vertex at the origin, with focus at \((0, -2)\) is \(y=-(1/8)x^2\)

Key Concepts

Standard Form of Equation

The parabola is a famous curve in mathematics, and its equation in standard form reveals a lot about its properties. The standard form for a parabola that opens either up or down is

• \((x-h)^2 = 4a(y-k)\)

where \((h, k)\) is the vertex of the parabola. This equation shows how the parabola is opened and oriented.
When the parabola opens left or right, the roles of \(x\) and \(y\) are switched:

• \((y-k)^2 = 4a(x-h)\)

The term \(4a\) determines how wide or narrow the parabola is. A larger absolute value of \(a\) means the parabola is narrower, while a smaller absolute value means it is wider.
Understanding the standard form helps us quickly find information about the parabola's direction, width, and position.

Vertex of Parabola

The vertex of a parabola is a critical point where the curve changes direction. In the standard form equation, \((h, k)\) is the vertex of the parabola. The vertex is essential because it provides the anchor point around which the parabola is shaped.
There are two primary cases for a parabola's vertex:

• For a parabola that opens up or down, the vertex is the point of minimum or maximum value, respectively.
• For a parabola that opens left or right, the vertex is the point of least or greatest \(x\)-value.

In this exercise, the vertex is at the origin \((0,0)\), indicating a symmetrical and centered parabola around this point. When graphing, starting from the vertex is helpful as it sets the stage for the shape and direction of the entire curve.

Focus of Parabola

The focus of a parabola is a special point located inside the parabola, along its axis of symmetry. The parabola's definition is that it is the set of all points equidistant from the directrix and the focus.
In our example, the focus is at \((0, -2)\), which indicates the parabola opens downwards in a vertical orientation. The value of \(a\) (from the standard form equation) is derived from the distance between the vertex and the focus. In this case, \(-2 - 0 = -2\), so \(a = -2\).

• This implies a negative \(a\) value, confirming the downward direction.
• The vertex to focus distance informs not just the direction but also the steepness of the parabola.

Knowing the focus and vertex allows precise plotting of the parabola, thus making calculations and drawing this curve straightforward.