1-8 Find the vertex, focus, and directrix of the parabola and sketch its graph. y^2+6 y+2 x+1=0

Name: Problem 7, Chapter 10: Calculus
Uploaded: 2021-06-16T20:13:13-08:00
Duration: 6 min 21 s
Channel: Tamara Worner
Description: 1-8 Find the vertex, focus, and directrix of the parabola and sketch its graph. y^2+6 y+2 x+1=0

1-8 Find the vertex, focus, and directrix of the parabola...

Key Concepts

Parabola

A parabola is a type of conic section defined as the set of points equidistant from a fixed point called the focus and a fixed line called the directrix. It has a distinct U-shaped curve and various algebraic representations depending on its orientation.

Vertex

The vertex of a parabola is the point where it turns; it is the midpoint between the focus and the directrix. In algebraic terms, rewriting the equation into vertex form through methods like completing the square reveals the vertex coordinates, which are crucial for graphing.

Focus

The focus is a specific point on the interior of the parabola such that every point on the parabola is equidistant from the focus and the directrix. Its position relative to the vertex helps determine the parabola's shape and orientation.

Directrix

The directrix is a fixed line used in the definition of a parabola, lying opposite the focus with the vertex exactly halfway in between. It, along with the focus, defines the set of points that constitute the parabola.

Completing the Square

Completing the square is an algebraic method used to transform a quadratic equation into a form that makes the vertex and other key geometric features of a conic section readily apparent. This technique is essential for rewriting the equation of a parabola into vertex form for analysis and graphing.

Transcript

00:01 Okay, for this question, the key concepts we're going to use is if our parabola is in the form y squared equals 4px, then the vertex is 0 ,0, the focus is p -0, and the directrix is x equals negative p.

00:19 So now in this problem, we have the equation of the parabola is y squared plus 6y plus 2x plus 1 equals 0.

00:29 So the first thing we have to do is get it to look more like this form.

00:35 And this parabola has some shifting going on.

00:37 So we're going to have to deal with that also.

00:40 But let's start working on getting it in the correct form.

00:43 First of all, i'm going to get all the ys together on one side and all the x's together on the other side.

00:50 So y squared plus the 6y equals negative 2x.

00:57 And then i'm going to subtract the one also.

01:01 The constant on the side with the x's.

01:05 All right.

01:06 Now on this left hand side with the y's, what i really want to do is i really want to complete the square so i can get something squared.

01:14 And again, if you guys remember how we complete the square, we're going to take half of the six, which would be three and square it.

01:24 That would get me nine.

01:28 But i can't just add nine on to one side of the equation.

01:32 I have to add it on to both sides of the equation.

01:36 And so again, here we are completing the square.

01:40 And we decided we wanted a nine there because then this left -hand side factors as into y -plus -3 quantity squared.

01:51 So again, the left -hand side becomes a y -plus -3 quantity squared.

01:56 The right -hand side becomes negative 2x plus 8.

02:05 And now we need to get that as to something times x.

02:11 So we're going to do y plus three squared.

02:15 Leave that here.

02:18 Equals negative 2 times parentheses x minus 4 if i factor out that negative of 2.

02:30 So now this tells me i have a shift going on.

02:33 Instead of my y's, i have a y plus 3, and instead of an x, i have an x minus 4.

02:41 Okay.

02:43 The other thing i need to do to get things going is i need to figure out what my p value is...

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