In Exercises 9-16, find the vertex, focus, and directrix of the parabola, and sketch its graph.

In Exercises 9-16, find the vertex, focus, and directrix of...

Key Concepts

Parabola

A parabola is the set of all points equidistant from a fixed point, called the focus, and a fixed line, called the directrix. It is a conic section that can result from slicing a cone with a plane parallel to its side. Understanding the parabolic shape is crucial for analyzing its geometric and algebraic properties.

Vertex

The vertex of a parabola is the point where the curve changes direction and is the point of minimum or maximum distance from the directrix and focus. It is often used as the reference point when writing the equation in its standard form, and it serves as the center of symmetry of the parabola.

Focus

The focus is a fixed point used in the geometric definition of a parabola. Every point on the parabola is equidistant from the focus and the corresponding point on the directrix. The position of the focus relative to the vertex determines the opening direction and the 'sharpness' of the parabola.

Directrix

The directrix is a fixed line that, in conjunction with the focus, defines the parabola. Any point on the parabola is exactly as far from the focus as it is from the directrix. This concept is fundamental in deriving the equations and properties of parabolic curves.

Standard Form of a Parabola

Writing the equation of a parabola in its standard form simplifies the identification of its key features such as the vertex, focus, and axis of symmetry. Converting to this form can reveal translations and dilations that have been applied to the basic parabola, and it aids in graphing and analysis.

Transcript

00:04 Hello, we're given an equation of a parabola which is x squared plus 6y equals 0.

00:20 So this is going to be a parabola.

00:25 So we will have a look at theorem 101.

00:29 Let's have a look at theorem 101 says.

00:32 It says that the standard formula for a parabola is when one of the variables is squared and the other is not.

00:45 So we see that the variable x is squared here and here the variable y is squared.

00:51 So this is not going to be the case.

00:54 We are going to focus on the case where there is a vertical axis.

01:00 Okay, theorem 10 tells us that the focus is below or above the the vertex the vertex v has coordinates hk okay so this vertex v it will the problem will open up or down depending on whether this parameter p in the standard equation is positive or negative okay so let's see what do we have here we have this is our formula it hasn't been written in the standard form so what we will do is we will subtract minus 6y to get x squared equals minus 6 y now we have we have the situation where the x is x minus 0 squared and this is minus 6 minus 6 times y minus 0 now this minus 6 we can write as we can write this minus 6 as we can write this minus 6 as it is negative it's negative but it's four times six over four okay four and four cancels and this is so this is our p it's negative p is equal minus six fourths or it's equal minus three halves so this is negative we will have we will have the situation where the parabola opens down.

02:43 It will open down.

02:46 We forget about this situation here.

02:51 And we also have found that the vertex is at h is equal 0 and k is equal 0.

03:00 Therefore the vertex is at 0.

03:03 We have found the vertex.

03:06 The focus, the focus is the focus is the focus is at a point h, okay, 0, 0 plus what is the p minus 3 half.

03:22 So 0 minus 1 .5...