Find the equation of the parabola described. Find the two points that define the latus rectum, a

Find the equation of the parabola described. Find the two...

Key Concepts

Parabola

A parabola is a set of points in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix. This geometric definition underpins the many properties of parabolas, including their reflective symmetry and characteristic U-shaped curves in standard forms.

Vertex

The vertex of a parabola is the point where the curve changes direction and is the closest (or farthest) point from the directrix. It plays a crucial role in the standard equations of a parabola and is often considered the point of symmetry for the graph.

Axis of Symmetry

The axis of symmetry is an imaginary line that divides the parabola into two mirror-image halves. For a parabola with a vertical orientation, this line is vertical, and it passes through the vertex. The axis of symmetry is fundamental in understanding the geometric structure and equation of the parabola.

Focus and Directrix

Every parabola is defined by its focus and directrix. The focus is a fixed point inside the parabola, and the directrix is a fixed line outside the parabola. The defining property of the parabola is that any point on the curve is equidistant from the focus and the directrix. This relationship is essential in deriving the parabola’s equation.

Latus Rectum

The latus rectum is a line segment that passes through the focus of the parabola, is perpendicular to the axis of symmetry, and has endpoints on the parabola. Its length, which is related to the parameter (typically denoted as p) in the parabola’s equation, provides important information about the parabola's width and geometry.

Standard Form Equation

The standard form of the equation of a parabola, especially one that opens vertically, is generally written as y = (1/(4p))x^2 or a similar variant in vertex form such as (x - h)^2 = 4p(y - k), where (h, k) is the vertex. This form directly highlights the relationship between the vertex, focus, latus rectum, and the overall shape of the parabola.

Graphing Techniques

Graphing a parabola involves plotting the vertex, determining and marking key points like the focus and the endpoints of the latus rectum, and drawing the curve that represents the set of all points satisfying the parabolic equation. Understanding how to identify and use these key elements is essential for an accurate representation of the parabola on the coordinate plane.

Transcript

00:01 So in this problem, we're given that the vertex of our parabola is at the origin, and our axis of symmetry is the y -axis, and that our parabola contains the point to three.

00:32 And we're asked to find the equation of the parabola, find the two points of the lattice rectum, and graph the equation.

00:42 So let's kind of begin with the graph here for a minute.

00:45 This is x and this is y.

00:47 We know the vertex is here.

00:51 We know the axis of symmetry is that y -axis.

00:55 And we know that this goes through the point two three, which means this parabola has to go pointed upward, doesn't it? in order to have this point right here at two, three on it.

01:12 Okay.

01:12 Since we know this problem is going upwards like that, then we know that the general formula is x minus h squared equals 4a times y minus k.

01:31 From our vertex being 0 ,0, we know h and k.

01:35 So that means i have x squared equals 4a y because h and k roll 0.

01:43 Okay, since i know it contains that point, i could plug that.

01:47 Point in here and solve this for a so 2 squared is 4 a times 3 so that means that 4 equals 4 a times 3 i divide both sides by 4 i have 1 equals 3 a so 1 3rd is now a so going back up to here that means my formula for this parabola is x squared equals 4 over 3 y because a is one third right okay so there's our formula our equation now to find the lattice rectum points remember these are the points on the curve with the line that goes through the focus okay and i know the focus is up here somewhere right has to be for the way that that's curved okay, so we know from the general formula that the formula for the focus, i'll mess that up, did not? let me rewrite that...

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