Problem 5 - Precalculus with Calculus Previews

Question

Find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \(x^{2}=-16 y\)

Step-by-Step Solution

Answer

Vertex: (0,0); Focus: (0,-4); Directrix: \(y=4\); Axis: \(x=0\).

Step 1: Identify the Parabola Form

The given equation is in the form of a vertical parabola. A standard form for a vertical parabola is either \(x^2 = 4py\) or \(x^2 = -4py\). In this case, the equation \(x^2 = -16y\) indicates it's in the form \(x^2 = -4py\), suggesting that the parabola opens downwards.

Step 2: Find the Vertex

For the equation \(x^2 = -4py\), the vertex is located at the origin, which is the point (0,0). This is derived directly from the standard form \(y = \frac{1}{4p}x^2\), where no other constants affect the vertex location.

Step 3: Determine the Value of p

From the equation \(x^2 = -16y\), we identify \(-4p = -16\). Solving for \(p\) gives \(p = 4\).

Step 4: Find the Focus

For a parabola in the form \(x^2 = -4py\), and since \(p = 4\), the focus lies at \((0, -p)\), which is the point (0, -4). This is because the focus is always \(p\) units away from the vertex along the axis of symmetry.

Step 5: Identify the Directrix

The directrix of the parabola is a line that is \(p\) units away from the vertex in the opposite direction of the focus. For this parabola, the directrix is the line \(y = 4\).

Step 6: Determine the Axis of Symmetry

The axis of symmetry for this parabola is the vertical line that passes through the vertex and is parallel to the y-axis. This line is \(x = 0\).

Step 7: Graph the Parabola

To graph the parabola, plot the vertex at (0, 0), the focus at (0, -4), and the directrix as the line \(y = 4\). The parabola will open downwards from the vertex, making sure the vertex is equidistant from the focus and the directrix.

Key Concepts

Vertex

The vertex of a parabola is a critical point where it changes direction. For the given parabola equation, \(x^2 = -16y\), you start by identifying the vertex. In a standard parabola with no translations, the vertex sits at the origin, (0,0). This is derived from the standard form of a parabola, \(x^2 = 4py\) or \(x^2 = -4py\), where any constants that might shift the vertex horizontally or vertically are absent. Here’s why it's so important:

• It serves as the starting point for graphing the parabola.
• It's equally spaced between the focus and the directrix.
• The vertex gives us insight into the maximum or minimum value of the parabola depending on its orientation.

Focus of a Parabola

The focus of a parabola is a special point located inside the curve. For the equation \(x^2 = -16y\), after identifying \(p = 4\), the focus can be found by locating a point that is \(p\) units away from the vertex along the axis of symmetry. Since our parabola opens downward, the focus will be at the point (0, -4). The focus is crucial to understanding how a parabola "behaves":

• Any line (called a "ray") drawn from the focus to the parabola will reflect off the curve and travel parallel to the axis of symmetry.
• The distance from any point on the parabola to the focus is equal to its distance from the directrix.
• This property of equilibrium between the focus and directrix defines the geometric nature of a parabola.

Directrix

The directrix of a parabola is an imaginary line that complements the focus. It's a fixed-line outside the parabola that helps set the curve's shape. To find it for the parabola \(x^2 = -16y\), you calculate and place it \(p\) units away from the vertex, but in the direction opposite of the focus. For our problem, the directrix is the line \(y = 4\). A closer look at its role:

• The directrix provides a reference line for maintaining the set distance balance with the focus.
• Every point on the parabola is equidistant from the focus and the directrix.
• This balance is what forms the characteristic "U" shape of the parabola.

Axis of Symmetry

The axis of symmetry is a vertical line that divides a parabola into two mirror-image halves. For the parabola described by \(x^2 = -16y\), this line is expressed as \(x = 0\). It runs vertically through the vertex and is parallel to the y-axis. Understanding the axis of symmetry helps in discovering the parabola’s symmetry and orientation.Here are a few key points:

• It’s crucial for ensuring the parabola’s shape is symmetrical.
• Any point on one side of the axis has a mirrored point on the other side.
• This line helps in sketching and understanding the proportional layout of the parabola.