Problem 3 - 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. $$x^{2}=-12 y$$

Step-by-Step Solution

Answer

Focus is at (0, -3); directrix is y = 3.

Step 1: Identify the Standard Form

A parabola is given in the form \(x^2 = -12y\), which indicates it is a vertical parabola (opening downwards). It matches the standard form \(x^2 = 4py\).

Step 2: Determine the Value of p

Compare \(x^2 = 4py\) with \(x^2 = -12y\) to find \(4p = -12\). Solve for \(p\): \(p = -3\).

Step 3: Find the Coordinates of the Focus

For a vertical parabola \(x^2 = 4py\), the focus is at the point \((0, p)\). With \(p = -3\), the focus of the parabola is at \((0, -3)\).

Step 4: Write the Equation of the Directrix

The equation of the directrix for \(x^2 = 4py\) is \(y = -p\). Substituting \(p = -3\) gives \(y = 3\) as the equation of the directrix.

Step 5: Sketch the Parabola

Draw the parabola opening downwards on the coordinate plane, ensuring the vertex is at the origin \((0, 0)\), the focus is at \((0, -3)\), and the directrix is the line \(y = 3\). Label the focus and directrix on the sketch.

Key Concepts

Parabola

The parabola is a fascinating curve and is part of the conic sections, which are curves obtained by intersecting a cone with a plane. A parabola has a U-shape and can open upwards, downwards, or sideways, resembling a bowl. One of the key features of a parabola is that it is symmetric, meaning if you were to fold it through its vertical axis (axis of symmetry), both halves would match perfectly.

A great way to identify a parabola is by looking at its equation. Parabolas can be written in the form of quadratic equations, often seen as either \( y = ax^2 + bx + c \) or \( x = ay^2 + by + c \). This standard form helps in determining the direction in which the parabola opens:

• If the parabola is of the form \(y = ax^2\), it opens either upwards (when \(a > 0\)) or downwards (when \(a < 0\)).
• If the parabola is of the form \(x = ay^2\), it opens left or right, depending on the value of \(a\).

Each parabola has a vertex, which is its highest or lowest point, and this vertex acts as a pivotal point for determining other features, like the focus and directrix.

Focus of Parabola

The focus is one of the pivotal characteristics of a parabola. It is a specific point inside the parabola which allows it to have its unique symmetric shape. The focus lies along the axis of symmetry, which is the line that divides the parabola into two identical halves.

For a vertically oriented parabola, represented by the equation \( x^2 = 4py \), the focus is located at the point \( (0, p) \).

• In our example \(x^2 = -12y\), we identified \(p = -3\).
• So, the focus is at \((0, -3)\).

This means if you were to draw a diagram of this parabola, the focus is positioned three units below the vertex. It's important because all the points on the parabola are equidistant from the focus and the directrix, a defining trait of parabola geometry.

Directrix of Parabola

The directrix of a parabola is an essential line that is external to the curve, working alongside the focus to define the structure of the parabola. It is essentially a line perpendicular to the axis of symmetry that maintains an equal distance from any point on the parabola to the focus.

For the vertical parabola given by \(x^2 = 4py\), the directrix can be determined by the equation \(y = -p\).In our specific example \(x^2 = -12y\), with \(p = -3\), the directrix is:

• \(y = 3\)

Understanding the relation between the focus and the directrix is key to comprehending how the points of the parabola are structured, with the focus and directrix at equal distances from the curve. On a graph, you'll find the directrix parallel to the x-axis, three units above the vertex at \((0, 0)\).

Quadratic Equations

The term quadratic equation might sound complex, but it simply refers to a polynomial equation of the second degree. Typically, it involves the variable raised to the power of two. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, with \(a eq 0\).

These equations are crucial in defining parabolas. For instance, in the equation \(x^2 = -12y\), rearranging terms and identifying the constant that relates the squared term with the linear term helps in determining the nature of the parabola—in this case, its direction (downwards) and its defining properties like focus and directrix.

Quadratic equations can be utilized beyond just the equations of a parabola. They appear frequently in various branches of mathematics and science, assisting in modeling real-world phenomena, such as projectile motions, optimizing business problems, or in the financial sector for calculating profits and losses. Understanding them is fundamental to fully grasping more complex algebraic concepts.