The graph of each equation is a parabola. Find the vertex of the parabola and then graph it. See

step 1 Problem in the parabola, x is equal to 3y squared plus 6y plus 7. So in order to do this, we nee

The graph of each equation is a parabola. Find the vertex of...

Key Concepts

Graphing Quadratic Equations

Graphing quadratic equations involves plotting critical features such as the vertex, axis of symmetry, and direction of opening. Understanding these elements aids in accurately sketching the parabola and analyzing its behavior in a coordinate plane.

Completing the Square

Completing the square is a method used to transform a quadratic equation into vertex form. This technique makes it easier to identify the vertex of the parabola and understand the transformation from the standard form to the graph's key features.

Parabola

A parabola is the graph of a quadratic function, which is a conic section. It can open in different directions depending on the form of the equation. Recognizing that an equation represents a parabola is essential for understanding its geometric properties and behavior.

Vertex

The vertex is the highest or lowest point on a parabola, acting as a point of symmetry. This key point determines the location where the function reaches its maximum or minimum value and is crucial for graphing the quadratic function accurately.

Transcript

00:01 Problem in the parabola, x is equal to 3y squared plus 6y plus 7.

00:05 So in order to do this, we need to put it into this form here of x equals a, then in parentheses y minus k squared plus h.

00:12 As it sits right now, it is not in that form.

00:15 We have to do a process called complete in the square.

00:18 To begin, we move the 7 over to the left hand side.

00:21 So it goes x minus 7.

00:23 I'm going to leave a little space there, and now i'm left with 3y squared plus 6y.

00:28 From here, what we need to do is factor out the three from the six.

00:33 So it becomes three, then in parentheses, y squared plus two y, and then we'll leave a little bit of a space there.

00:40 The reason we do this is because we only are going to deal with b at this point, the 2y, and we can't have a number in front of the y squared.

00:47 So my next steps now are to do two divided by two.

00:52 So this number here, cut in half, and square it, which is equal to one square, which is, to 1.

00:59 So i'm going to add 1 over here.

01:02 Now on the left hand side, i got to make this balance.

01:05 Now i'm not just going to multiply 1, but i'm multiplied by the product of the number out front and 1.

01:10 So here is going to be 3 times 1.

01:14 So in the other words, i'm going to be doing x minus 7 plus 3.

01:18 So now i'm going to simplify this down a little bit.

01:20 The negative 7 and the positive 3 becomes x minus 4.

01:23 And on the right hand side, we're going to factor the y square plus 2y plus 1.

01:28 This factors into y plus 1 times y plus 1, or in other words, y plus 1 squared.

01:34 I'm going to finish by moving the 4 to the other side.

01:37 So i got 3 times y plus 1 square plus 4.

01:42 This is now in the correct form.

01:44 So now i can identify the vertex...

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