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Find an x^' y^'-equation for the graph of the given x...

Key Concepts

Parabolic Curves

Parabolic curves are a type of conic section characterized by a quadratic relationship between the variables. They exhibit a distinct symmetry about a line through the vertex, and their equations can be simplified via translation and completing the square, aiding in the identification of their key features.

Vertex Form of a Parabola

The vertex form of a parabola expresses the quadratic equation in the form y = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form makes it clear where the vertex is located and how the curve opens, which is critical for graphing and understanding its geometric properties.

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic equation into a perfect square plus a constant. This process simplifies the expression and exposes key features like the vertex or center of a conic section, making it easier to analyze and graph the curve.

Coordinate Translation

Coordinate translation is the method of shifting the coordinate system so that the new origin coincides with a significant point on the graph, such as the vertex for a parabola. This transformation simplifies the equation of the curve, revealing its standard form and symmetric properties.

Transcript

00:02 Hello, we're given an equation x squared plus 4x plus 4y minus 4 equals 0.

00:13 This is a parabola.

00:16 How can we tell that it's a parabola? one of the variables has a square term and the other does not.

00:23 So there are now mixed...

00:26 This is of the three possibilities, this is a parabola.

00:33 If both variables had there if both variables have terms with squares of them then the curve would be either an ellipse or a hyperbola or a special type of ellipse which is a circle okay so we are working with parabolas if we're working with parabolas we will be probably theorem 101.

01:06 Now what type of a parabola is it? is it this type or this type here? even though it hasn't yet been written we need to write this in standard form.

01:17 But we see that with the vertical axis the x term has been squared and the horizontal axis parabola the y term has been squared.

01:26 Which terms do we have squared here? this one, right? so we are going to going to have a parabola with a vertical axis.

01:38 Okay, it's a vertical axis.

01:43 When we write, if we succeed to write this in the standard form, we will be able to read what the vertex coordinates are.

01:55 We will be able to find out what p is, depending on p.

01:59 We'll see which side it opens, does it open up or down? and then if this is the vertex for example if this is the vertex then the distance is p between the vertex and the focus and also it's p to the directrix the directrix it's going to be horizontal line p units below v if it opens up and if it opens down then it's the other way around if is p units below the vertex and the directory is pinned above the vertex.

02:41 So let's see if we can work this out.

02:44 So we know that the x terms are going to be squared, so anything that is not an x term, these two will travel to the other side.

02:53 We are going to subtract 4y and we're going to add 4.

02:59 And what we are left with on the left hand side, is this.

03:04 What we're doing now is completing the square.

03:13 So we want to add, we want to add something to either side.

03:21 So this makes the square of x plus something.

03:30 Let's see.

03:31 If i have x plus a square then i will have the first term squared, i'll have the second term squared, and i will have plus 2a x.

03:46 So my 2a is 4, therefore a is equal to.

03:53 And i'm going to add 2 squared to both sides.

04:00 So i can write that this is x plus 2 squared equals, and this is minus 4, 1.

04:11 Plus 4 plus 4 which is minus 4 y plus 4 plus 4 and plus 4 is plus 8 we have succeeded in writing the left hand side of the standard equation as it should be now we draw our attention to the right side what can we do with this we can factor out we can factor out minus 4 and we're left with y plus 2 so we have x plus 2 squared and this is now written in the form that we want form here.

04:56 From this we will gather that 4p is equal to minus 4.

05:02 Therefore p is equal to minus 1.

05:06 When p is equal to minus 1 we see that it's negative.

05:11 Since it's negative, this is the situation that's going to happen.

05:16 Okay so we will eliminate all these other situations furthermore this is x minus h and in these parentheses we have y minus k so h is equal to minus two and k is equal to minus two and k is is equal to minus two so our vertex our vertex is at minus 2, minus 2, okay? and our p is equal minus 1.

06:15 So if this vertex is at minus 2, minus 2, then one unit below it at minus 2, minus 3, we are going to have the focus, and one unit above it, which is minus 1, y equals minus one we will have the directrix and there we have the vertex the focus and the directrix we have figured them out and written them here okay now we will sketch the situation sketching the situation i'll do it here first crudely by hand and then i'll bring a better sketch from a program.

07:04 Okay so minus two minus two minus one minus two minus two minus one minus two our vertex is here the parabola opens down so through this vertex we're going to draw through the vertex we will draw a vertical line and a horizontal line through the vertex we will have a horizontal and a vertical line which we usually are with dashed lines because they're just auxiliary lines here.

07:45 Right.

07:47 So this was our x -axis...

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