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

Graphing a Parabola

Graphing a parabola involves plotting essential features such as the vertex, focus, and directrix, along with sketching the curve that reflects the parabola’s symmetry and orientation. This process provides a visual representation of the parabola and supports deeper insights into its geometric and functional properties.

Directrix

The directrix of a parabola is a fixed straight line used in the geometric definition of the curve. The distance from any point on the parabola to the directrix is equal to the distance from that point to the focus. This concept is integral to both the theoretical understanding and practical plotting of parabolic graphs.

Standard Form Conversion (Completing the Square)

Converting a quadratic equation into the standard form of a parabola often requires the technique of completing the square. This manipulation not only simplifies the equation but also makes it easier to extract key information such as the vertex, orientation, and the values relevant to calculating the focus and directrix.

Vertex

The vertex of a parabola is the point where the curve attains its maximum or minimum value; it is the turning point of the graph. In the standard form of a parabola, the vertex is easily identifiable, and knowing its coordinates is crucial for accurately graphing and analyzing the parabola's properties.

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. It has a distinct U-shaped curve and exhibits reflective symmetry along its axis, making it significant both in theoretical mathematics and practical applications such as physics and engineering.

Focus

The focus of a parabola is a fixed point such that any point on the parabola is equidistant from the focus and the directrix. This unique property of parabolas is used in deriving the curve’s equation in its standard form and is important for understanding the reflective characteristics of the parabola.

Transcript

00:02 Hello, we are given the equation y squared minus 4 x is equal 0 and we know that this is a parabola because one of the one of the variables is squared and one is not one does not have a square term so since we're talking about a parabola parabola we refer to theorem 101 that talks about parabolas.

00:46 Let's put it here in this one here, right? so this has not yet been written in standard form, but we can say which one of these is going to be applicable here.

01:02 Is the term containing x squared or the one containing y? the one containing y so it's going to have a horizontal axis which means that we can eliminate this situation here right we're going to have a parabola with a horizontal axis it's either going to open left or right once we write it in standard form we'll find it's h and k for the vertex and then the focus will be to the left or to the right by p units and the directrix will be a vertical line p units to the left or to the right depending on where this parabola opens who now let's see what we can do we can write this as we can write this we're going to attempt to write this in the standard form so if we take the all the term containing y squared minus 4 y we are going to and transfer this to the other side we'll add 4x to the other side say this 4x right what do we need what do we need to add what do we need to add here what do we need to add here to make this a perfect square so look if we have two terms that are subtracted and squared we will have the square of the first plus the square of the second minus double the product of the two.

02:49 So this would be two times the first which is y times what? this is two times two times two, right? so our second term is two which means we add two squared and we take away two squared.

03:08 Because we don't want to change the value of the expression on the left -hand side.

03:14 Now we interpret, we recognize this to be y -minus 2 squared.

03:22 This is called completing the square.

03:30 Okay, and what do we do with this term? we added to both sides, so it comes to this side here.

03:43 We have the left side covered.

03:45 Now we need to show, we need to factor this.

03:49 So this is going to be four, we factor out four and x plus one.

03:53 And there we have our standard form as we wanted it, as it was listed in theorem four.

04:03 Now, this is y minus k.

04:08 This is x minus h.

04:10 So h is equal minus one and k is equal to two.

04:16 4 is 4 is equal to 4 p.

04:22 See 4p? so dividing by 4 we have p equals 1.

04:33 What are we going to have? p is positive.

04:37 It's greater than 0.

04:38 So this is the situation we will have.

04:41 We forget about the one that turns to the left.

04:44 So, if i have the vertex, the vertex is going to be at, where do we say, hk is minus 1, minus 1 2.

04:58 I am going to plot minus 1 2 as my vertex and through that, through that vertex, i will draw a vertical and a horizontal line, lines that are parallel to the axes...

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