Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. $$ x^{2}=-16 y $$

Step-by-Step Solution

Answer

The focus is at (0, -4) and the directrix is y = 4.

Step 1: Identify the Standard Parabola Form

First, identify the standard form of the parabola equation. Here the equation is given as \( x^2 = -16y \). This aligns with the form \( x^2 = -4py \), indicating that the parabola opens downwards.

Step 2: Match to Find Value of p

Compare \(-16y \) in the given equation with \(-4py \) in the standard form. This gives \(-4p = -16\). Solving for \(p\), we find \(p = 4\).

Step 3: Determine Focus Coordinates

For a parabola \( x^2 = -4py \), the focus is at \((0, -p)\). Since \(p = 4\), the focus is at \((0, -4)\).

Step 4: Determine Directrix Equation

The equation of the directrix for the parabola \( x^2 = -4py \) is given by \( y = p \). Thus, for this parabola, the directrix is \( y = 4 \).

Step 5: Sketch the Parabola

Draw the parabola on a coordinate plane. The parabola opens downwards from the origin \((0,0)\). Mark the focus at \((0, -4)\) below the origin, and draw the horizontal line \( y = 4 \) above the origin as the directrix, clearly showing the relationship between the parabola, focus, and directrix.

Key Concepts

Focus of a parabola

The focus of a parabola is a special point that plays a significant role in defining its shape and direction. It is positioned on the interior of the parabola. The distance between any point on the parabola and its focus is equal to the perpendicular distance from that point to the directrix.
For parabolas with the equation of the form

• Opening upwards or downwards: \[x^2 = 4py\] The focus is at \((0, p)\) when opening upwards or \((0, -p)\) when opening downwards.
• Opening sideways: \[y^2 = 4px\] The focus is at \((p, 0)\) when opening to the right, or \((-p, 0)\) when opening to the left.

In the exercise given, the parabola is described by \(x^2 = -16y\), which fits the second case with\(-4py\). By comparing, we find \(p=4\). Thus, the focus is located at \((0, -4)\), an essential point that helps sketch and understand the curve's behavior.

Directrix of a parabola

The directrix of a parabola is a line that, together with the focus, helps define the parabola. It is located outside the curve on the opposite side of the focus. Any point on a parabola is equidistant from its focus and its directrix.
To find the directrix:

• For parabolas opening upwards or downwards like \[x^2 = 4py\] or \[-4py\], the directrix is horizontal, given by \(y = -p\) for upwards and \(y = p\) for downwards.
• For parabolas opening sideways like \[y^2 = 4px\] or \[-4px\], the directrix is vertical, given by \(x = -p\) or \(x = p\).

In our equation \(x^2 = -16y\), the directrix is \(y = 4\) because \(p=4\). This line is located above the vertex \((0,0)\), playing a crucial role in the geometric definition of the parabola.

Standard form of a parabola

The standard form of a parabola's equation provides a convenient way to quickly identify its key properties such as the direction, size, and orientation. The standard forms include:

• Upwards or downwards opening: \[x^2 = 4py\]
• Sideways opening: \[y^2 = 4px\]

The sign of \(p\) indicates the direction of opening — positive opens upwards/right and negative opens downwards/left. Additionally, the absolute value of \(p\) determines how broad or narrow the parabola is. Larger \(|p|\) values mean a wider parabola.
In our exercise \(x^2 = -16y\), this equation is a negative variant matching \(-4py\), thus opening downwards with \(p=4\). Recognizing this connection to the standard form allows easy determination of the focus and directrix.