Week 10, Day 1

Today we introduced conics, a term that encompasses all shapes formed by an intersection of a plane and a double-napped cone, including circles, ellipses, parabolas, and hyperbolas. This lesson focuses specifically on porabolas, a set of all points (x,y) that are equidistant from a fixed line (directrix) and a fixed  point (focus) not on the line. The midpoint between the focus and the directrix is called the vertex, and the line passing through the focus and the vertex is called the axis of the parabola. Using the definition of a parabola, you can derive the following standard form of the equation of a parabola whose directrix is parallel to the x-axis or to the y-axis. The vertex at (h,k), the standard form is: (x-h)^2 = 4p(y-k): vertical axis; directrix: y = k-p. (y-k)^2 = 4p(x-h): horizontal axis; directrix: x = h-p. The focus lies on the axis p units (directed distance) from the vertex. If the vertex is at the origin (0,0), the equation takes one of the following forms. x^2 = 4py: vertical axis. y^2 = 4px: horizontal axis. 

Example 1: Finding the Standard Equation of a Parabola

Find the standard form of the equation of the parabola with vertex (2,1) and focus (2,4).

Because the axis of the parabola is vertical, consider the equation (x-h)^2 = 4p(y-k) where h = 2, k = 1, and p = 4-1 = 3. Thus, the standard form is (x-2)^2 = 12(y-1). 

Example 2: Finding the Focus of a Parabola

Find the focus of the parabola given by y = -1/2(x)^2 - x + 1/2. To find the focus, convert to standard form by completing the square. The steps are listed as follows:

y = -1/2(x)^2 - x + 1/2 (Original equation)

-2y = x^2 + 2x - 1 (Multiply both sides by -2)

1 - 2y = x^2 + 2x (Group the terms)

2 - 2y = x^2 + 2x + 1 (Complete the square)

-2(y - 1) = (x + 1)^2 (Standard form)

Comparing this equation with (x - h)^2 = 4p(y - k), you can conclude that h = -1, k = 1, and p = -1/2. Because p is negative, the parabola opens downward. The focus is (h, k + p) = (-1, 1/2). 

Example 3: Vertex at the Origin

Find the standard equation of the parabola with vertex at the origin and focus (2,0). The axis  of the parabola is horizontal, passing through (0,0) and (2,0). Thus, the standard form is y^2 = 4px where h = k = 0 and p = 2. The equation is y^2 = 8x.