Sketching a Parabola In Exercises 11-16, find the vertex, focus, and directrix of the parabola,

Sketching a Parabola In Exercises 11-16, find the vertex,...

Key Concepts

Parabola

A parabola is a type of conic section defined as the set of all points in a plane that are equidistant from a fixed point called the focus and a fixed line known as the directrix. This unique distance property gives the parabola its characteristic ‘U’ shape and symmetry.

Vertex

The vertex is the point on the parabola where it changes direction, representing the minimum or maximum point. It is the point of intersection between the parabola and its axis of symmetry, and it plays a key role in graphing and analyzing the parabola’s shape and position.

Focus

The focus is a fixed point located inside the parabola along its axis of symmetry. It, together with the directrix, defines the parabola; every point on the parabola is equidistant from the focus and the directrix, which is central to the geometric definition of a parabola.

Directrix

The directrix is a fixed line associated with a parabola. For any point on the parabola, the distance to the directrix is equal to the distance to the focus. This line serves as a reference in the parabola’s definition and helps determine its orientation and position in the coordinate plane.

Completing the Square

Completing the square is an algebraic technique used to rewrite a quadratic equation into its vertex form. This process simplifies the identification of the vertex and other key features of a parabola, making it easier to analyze and graph the conic section.

Transcript

00:01 To find the vertex, focus, and directives of this problem, we're first going to rewrite it into standard form.

00:08 So let's begin by writing down the equation to x squared minus 2x, minus 4y minus 7, equals 0.

00:21 And to rewrite it into standard form, we're going to need to complete the square because there's a perfect square in the standard form of the parabola.

00:29 So we can complete the square in this part, x squared minus 2x, and the number we need to add is going to be half of negative 2 squared.

00:41 So half of negative 2 is negative 1.

00:43 Negative 1 squared is positive 1, so plus 1.

00:47 Because we're adding 1, to keep the equation unchanged, we're also going to subtract 1, and we'll still have the minus 4y minus 7.

00:59 All right.

01:01 So next we can factor this part, complete the square here.

01:04 This is factors into x minus 1 squared.

01:10 Next must combine this negative 1 here.

01:12 It's minus 1, which is minus 7.

01:14 We've still got minus 4y, but we're going to combine those two to make minus 8 and equal 0.

01:23 And next in standard form, we're going to get the x minus 1 squared by itself.

01:28 We're going to move everything else over to the right side.

01:31 So we'll have x minus 1 squared equals 4y plus 8.

01:40 And finally, i'm just going to factor out the 4, so that we'll have it in the form y minus something.

01:48 So it would be y plus 2.

01:50 That's right, yeah.

01:52 Now it's written as 1 squared equals 4 times y plus 2.

01:58 We can see the vertex right away...

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