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

Completing the Square

Completing the square is an algebraic method used to convert a quadratic equation into a perfect square trinomial, which simplifies rewriting the equation in standard form. This process is essential for identifying the vertex of a parabola and for revealing other key properties of the graph.

Directrix

The directrix is a fixed line used in the definition of a parabola, serving as a counterpart to the focus. It ensures that every point on the parabola has the same distance to it as to the focus, thereby influencing the shape and orientation of the parabola.

Standard Form of a Parabola

The standard form of a parabola's equation, typically expressed as (y - k)² = 4p(x - h) for a horizontally oriented parabola, clearly displays its geometric features. In this form, (h, k) is the vertex, and the parameter p relates directly to the distance between the vertex and the focus as well as the vertex and the directrix.

Vertex

The vertex is the point on the parabola where it turns; that is, it is the point where the parabola attains its minimum or maximum value, depending on its orientation. The vertex serves as a key reference point when rewriting the parabola’s equation in standard form.

Parabola

A parabola is a type of conic section defined as the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. This conic has a characteristic U-shape and is significant in various fields including physics and engineering due to its reflective properties and structural behavior.

Focus

The focus is a fixed point located inside the parabola. It, along with the directrix, defines the parabola since every point on the parabola is equidistant from the focus and the directrix. This property is fundamental to the geometric definition of a parabola.

Transcript

00:01 To find the vertex, focus, and directrix of this parabola, the problem we have is y squared plus 6y plus 8x plus 25 equals 0.

00:18 We're first going to rewrite it into standard form and that will help us identify the vertex as well as figure out what the directrix and focus are.

00:29 To rewrite this in standard form, we need to complete the square on the y squared term.

00:35 So let's do that over here.

00:37 We have y squared from 6 y.

00:40 To complete the square, the number we need to add is 6 divided by 3.

00:45 Divided by 2 is 3, 3 squared is 9.

00:48 We're going to be adding a 9.

00:51 And because we add a 9, we also need to subtract a 9.

00:54 So plus 9 minus 9, we've still got this 8x here, plus 25.

01:01 Equals 0.

01:07 Now we can factor this part, finish completing the square.

01:10 This part becomes y plus 3 squared.

01:18 That's this part here.

01:21 Over here, and let's combine this minus 9 to 25 over there to make plus 8x plus 16.

01:31 And finally, in the standard form of a parabola, the square term is on its own.

01:38 So to get this part by itself, let's move everything over to the right side.

01:41 And on the left side we're going to have y plus 3 squared equals negative 8 x minus 16.

01:53 And finally, final final step, to identify, to get it really in standard form, we need to factor out so that just x minus h.

02:04 Let's factor out the negative 8.

02:09 Now we'll have an x plus 2 length over.

02:13 And now that it's in standard form, you can identify the vertex.

02:19 This is x minus h this is y minus k so the vertex is at negative to negative 3 and we need a couple more pieces of information the first thing is what is p well this number in front of the x minus h is 4p so if negative 8 equals 4p in other words if negative 8 equals 4p this means that p what's p i could be you have to be you have to be be negative 1 half, excuse me, negative 2.

03:09 So, here's that, it's turning to a negative, yeah, negative 2, sorry.

03:17 So that's p, and a couple more things to notice, because this equation, this equation y plus 3 squared, y is being squared, the equation is going to open left and right, or the rather problem is going to open left and right...

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