Problem 40 - Precalculus Mathematics for Calculus

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focal diameter 8 and focus on the negative \(y\) -axis

Step-by-Step Solution

Answer

The equation of the parabola is \( x^2 = 8y \).

Step 1: Understand the Parabola's Orientation

Since the focus is on the negative y-axis, the parabola opens downwards. This means the equation of the parabola is of the form \( x^2 = -4py \).

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

The focal diameter is the absolute value of \( 4p \). We are given that the focal diameter is 8, so\[ |4p| = 8 \].Solving for \( p \), we divide both sides by 4:\[ |p| = 2 \].Since the parabola opens downwards, \( p \) is negative. Thus, \( p = -2 \).

Step 3: Write the Equation of the Parabola

Now that we know \( p = -2 \), substitute this value into the equation form:\[ x^2 = -4(-2)y \].Simplify the equation:\[ x^2 = 8y \].

Key Concepts

Vertex at Origin

A parabola can be thought of as a U-shaped curve that has particular points of interest, including the vertex. The vertex is the highest or lowest point of the parabola, depending on whether it opens upwards or downwards. In this case, the parabola in question has its vertex at the origin, which is the point

• (0,0)

. This means the axis of symmetry of the parabola passes through this point.

Having the vertex at the origin simplifies the equation of the parabola because it removes the need for translating the formula. Normally, if the vertex was at a different location, say

• (h,k)

, the equation would incorporate these values. Here, we focus only on the pure quadratic aspect of the parabola without any added constants for translation.

Focal Diameter

The focal diameter is a significant concept when working with parabolas. It refers to the total width of the parabola at the focus.

Mathematically, the focal diameter is given by the formula

• |4p|

, where

• p

is the distance between the vertex and the focus of the parabola. In our specific problem, the focal diameter is given as 8. To find

• p

, we solve

• |4p| = 8

. By dividing both sides by 4, we find that

• |p| = 2

. Knowing the direction of the parabola is vital as well, here it affects the sign of

• p

, providing us with

• p = -2

. This negative value indicates that the parabola opens in the negative direction along the y-axis.

Negative Y-Axis Focus

The focus of a parabola is a fundamental feature. It is a point from which distances are measured to define the parabola. For a parabola with the vertex at the origin opening downwards, the focus is located on the negative y-axis.

In our example, given that the vertex is

• (0,0)

and

• p = -2

, the focus would be at the point

• (0,-2)

. This position tells us the specific location below the vertex where the parabola is directed. Having this knowledge allows us to verify the nature of the equation derived, as it must match the orientation and size sketched by the parameters of focus and focal diameter. These checks ensure that the equation

• x^2 = 8y

provides an accurate representation of the described geometric shape.

The negative sign in the equation affirms the direction towards the negative y-axis, making all concepts harmoniously tie together.