Problem 35 - Algebra and Trigonometry

Question

\(29-40\) Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix: \(y=-10\)

Step-by-Step Solution

Answer

The equation of the parabola is \(x^2 = 40y\).

Step 1: Identify the Parabola Type and Orientation

Given that the directrix is a horizontal line at \(y = -10\) and the vertex is at the origin \((0,0)\), the parabola opens upwards. Parabolas that open up or down have the general equation form \(x^2 = 4py\).

Step 2: Determine the Value of \(p\)

The distance \(p\) from the vertex to the directrix is 10 units, since the directrix is at \(y = -10\) and the vertex is at \(y = 0\). Hence, \(p = 10\). Since the parabola opens upwards, \(p\) remains positive.

Step 3: Write the Equation of the Parabola

Using the general form \(x^2 = 4py\) and substituting \(p = 10\), the equation of the parabola becomes \(x^2 = 4(10)y\). Simplifying this gives \(x^2 = 40y\).

Key Concepts

Vertex Form of a Parabola

Understanding the vertex form of a parabola is key to easily identifying the parabola's properties from its equation. The vertex form of a parabola is expressed as \[ y = a(x - h)^2 + k \] where

• \( (h, k) \) is the vertex of the parabola,
• \( a \) determines the width and direction of the parabola's opening.

In the given exercise, since the vertex is at the origin \( (0, 0) \), the vertex form simplifies to \[ y = ax^2 \]. For parabolas that open upwards or downwards, the vertex form can be slightly modified into standard form \( x^2 = 4py \) where \( p \) is the distance from the vertex to the focus or the directrix. Incorporating this into a real-life example, if you toss a ball, its path forms a parabola with the vertex being the highest point of the throw. Identifying the vertex helps determine crucial aspects of the parabola, like its orientation and direction, which ties into the next concepts.

Directrix of a Parabola

The directrix of a parabola serves as a fundamental component in understanding its shape and location. A directrix is a line, which, together with the focus, gives a geometric definition to the parabola.For a parabola, every point on it is equidistant to a fixed point known as the focus and a fixed line known as the directrix. This means the parabola is the set of all points such that the distance to the focus equals the distance to the directrix.In our exercise example, the directrix is at \( y = -10 \), which communicates that the parabola does not stretch infinitely vertically; rather, it stops once it reaches the same distance from the origin as the distance to the directrix. From given conditions, we identify the directrix to correctly position the parabola in the space, making it possible to write the parabola's equation correctly.

Orientation of Parabolas

The orientation of a parabola tells us the direction in which it opens, and it is determined by both its vertex form and the sign of the parameter \( p \). In the equation \( x^2 = 4py \), \( p \) plays a critical role:

• If \( p > 0 \), the parabola opens upwards.
• If \( p < 0 \), it opens downwards.

For parabolas that open upward or downward, this orientation is fundamental for graphing and solving problems. In the given exercise, because the directrix lies below the vertex at \( y = -10 \), and \( p = 10 \), the parabola opens upwards. Understanding this helps in visualizing and analyzing the curve's path.When observing natural phenomena like the path of projectiles or satellite dishes, recognizing the parabola's orientation enhances our comprehension of their behaviors and pathways.