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

Directrix

The directrix is a fixed line that, along with the focus, defines a parabola. The property that a parabola consists of all points equidistant from the focus and the directrix is pivotal in both its definition and construction.

Graphing a Parabola

Graphing a parabola involves plotting the vertex, focus, and directrix, and then sketching a curve that reflects its symmetry about the axis of symmetry. Understanding these components through the standard form enables one to accurately depict the upward, downward, leftward, or rightward opening of the parabola.

Focus

The focus of a parabola is a fixed point located inside the curve, which, together with the directrix, defines the parabola through its distance properties. Every point on the parabola is equidistant from the focus and the directrix.

Standard Form of a Parabola

The standard form of a parabola’s equation, such as (x - h) = a(y - k)^2 or (y - k) = a(x - h)^2, makes it easier to identify the vertex (h, k) and understand the parabola's orientation and width. This form is crucial in analyzing and graphing parabolas.

Vertex

The vertex of a parabola is the point where the curve changes direction; it is the minimum or maximum point of the parabola. In the standard form, the vertex is given by the coordinates (h, k), and it serves as a key reference for determining the properties of the parabola.

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic expression or equation into a perfect square plus a constant. In the context of conic sections, it is essential for rewriting the equation of a parabola into its standard form, which clearly reveals the vertex and other geometric properties.

Parabola

A parabola is a conic section defined as the set of all points that are equidistant from a fixed point called the focus and a fixed line called the directrix. This geometric shape has a distinct U-shaped curve and appears in various branches of mathematics and physics.

Transcript

00:01 Hello, we are given a formula an equation y squared y squared plus 6y minus plus 8x plus 95 is equal to 0 this is a parabola because one of the one of the terms is squared one of the terms is squared one of the the variables is squared and the other has no square term so this will be a parabola will find that soon enough why and how but for parabolas when we work with parabolas we turn our attention to theorem 101 so theorem 101 talks about talks about and their equations okay let's see let's see two determine which type of a parabola this is going to be.

01:14 We can tell this straight away because we can ask ourselves is the x term squared? let's see, do we have square of an x term? no we don't.

01:22 Do we have y is going to be eventually squared here? yes, so this one is going to be a horizontal.

01:30 This case is going to be.

01:32 So the vertical axis story we can forget about for now, okay? it's going to be a parable with a horizontal axis, it's going to have some vertex with hk coordinates, it's either going to open left or right, depending on whether this p, this p here is positive or negative, and when we find the hk and the p, we will find where the focus is, and when we find with the focus is, that then the focus is, if the focus is to the left and the directrix will be to the right and if the focus is to the right of the vertex, the directrix will be to the left.

02:21 Let's go and see what we can do with this equation that we are given.

02:29 We are going to take anything that is not connected to y and shift it to the other side.

02:39 Side.

02:40 So we will take away 8x and take away 25.

02:45 Alright, so we will have minus 8x minus 25 here.

02:51 And here we will say what do we need to add and subtract in order for this to be a complete square? so if we have, if we have something plus something, something else.

03:13 I'm saying plus because this term has plus in front of it.

03:17 Then expanding the square of a sum gives us the first term squared, we have the second term squared and we have the double of the product of the first and second term.

03:33 So what can we say here? the first term is y.

03:37 And this six is two times y.

03:41 Times 3 right so we have y plus 3 squared if we have y plus 3 squared then we will have y squared plus 2 times y times 3 is 6 squared and 6 times right and we need to have 9 here so in order to make this expression a total square we add 9 and subtract.

04:16 This method is called completing the square.

04:23 The square.

04:25 So we have the square of y plus 3 squared.

04:30 Don't forget this.

04:32 What's this 9? it's 3 squared, right? so we have minus 9 equals minus 8, x minus 25.

04:42 In order for us to get a standard this is a standard equation.

04:51 This minus 9 is in the way.

04:55 So we add 9 to both sides.

04:58 And we have y plus 3 squared equals minus 8x minus 16.

05:08 Okay, the left hand side is good now.

05:13 What do we do with the right hand side? it needs to be factorized.

05:17 So what can we factor out here? we can factor minus 8 out and we're left with x plus 2 minus 8 times x is minus x minus x minus 8 times plus 2 is minus 16 good is the standard form of the equation yes it is we just need to work out what this business with 4p is so we have this minus 8 is 4p is minus 8.

05:58 So p is minus 8 divided by 4 is minus 2.

06:02 Okay, this is negative.

06:07 So when we have negative p, we have this situation.

06:12 It opens left.

06:16 And this situation can be discovered now...

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