(a) Find equations for the family of parabolas with vertex at the origin, focus on the positive

(a) Find equations for the family of parabolas with vertex...

Key Concepts

Focal Diameter (Latus Rectum)

The focal diameter, also known as the latus rectum, is the chord that passes through the focus and is perpendicular to the axis of symmetry. Its length is given by |4p| and plays a key role in characterizing the width of the parabola, directly affecting the shape of the curve.

Family of Parabolas

Considering a family of parabolas involves examining how specific parameters, such as the focal diameter or the focal length p, vary. In such families, although the general shape remains a parabola, differences in these parameters result in variations in the width and steepness of the curves.

Standard Form of an Upward-Opening Parabola

A parabola with its vertex at the origin and opening upward has the standard form x² = 4py, where p is the distance from the vertex to the focus. This equation encapsulates the geometric definition of a parabola as the set of all points equidistant from the focus and the directrix.

Focus and Vertex Relationship

The vertex of a parabola is the point where the curve changes direction, and for a parabola opening upward with its vertex at the origin, the focus lies on the positive y-axis at the point (0, p). This relationship is crucial for determining the specific equation of a parabola based on the position of its focus.

Graphical Behavior and Scaling

Graphical analysis of a family of parabolas with varying focal diameters demonstrates the effects of changing p on the spread and orientation of each curve. As the focal diameter increases, the value of p increases, leading to parabolas that open more widely. This concept is fundamental to understanding transformations and scaling in conic sections.

Transcript

00:01 All right, so we have various parabolas with focal diameters of 1, 2, 4, and 8.

00:07 We each defines their equations.

00:11 And we also know that our fosy for these parabolas are above the x -axis, and they're on the y -axis.

00:19 So at 0 -p, where p is positive.

00:23 So that means that our equations are going to take the form of x squared equals 4 times p times y, where p is just going to be the focal diameter.

00:32 So we plug in 1, 2, 4, and 8 into p here.

00:36 We will get x squared equals, i'm just going to write like this.

00:45 When the diameter is 1, x squared equals 4 y, when 2 it's 8y, 1, 4 it's 16 y, and then finally 32 y.

01:03 And if we want to rewrite these in the form y equals of something we can just divide both sides by four eight 16 or 32 and for the first one we'll have let's do this y equals x squared over 4 for the next one x squared over 8 and x squared over 16 and x squared over 32 all right now all that's left is to draw these.

01:48 So let's start with the first one.

01:49 Y equals x squared over four.

01:51 I'm going to use black for this.

01:54 We start in 0 .0.

01:56 We know that our vertex is at 0.

02:00 Then we can plug in various points and see what the value is going to be there.

02:04 So for example, when x equals 1, we plug that in, y equals 1 squared over 4, so just 1 4th.

02:14 So approximately here.

02:17 And when negative 1, negative 1 squared is the same as 1 squared, so it's going to be symmetrical.

02:23 Well, that's roughly symmetrical.

02:26 Okay.

02:26 When we plug in 2, 2 squared is 4.

02:30 4 over 4 is 1.

02:32 So like this.

02:36 And when we plug in 3, or rather 4 is probably going to be easier to plug in 4 squared over 4 is just 4.

02:44 So, and this allows us to get a loose outline for the shape of our platform.

02:52 And that was a lot more messy on the left side, but whatever.

03:04 Then we move on to the second one.

03:05 We do the exact same thing.

03:06 So once again, we start at 0 -0.

03:11 And then if we plug in, for example, 2, 1 is going to be difficult to draw because we're going to end up with 1 8.

03:20 So let's plug in 2 instead.

03:23 2 squared over 8 is 1 half.

03:28 Maybe 2 squared over 8 is the same thing...

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