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

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

Key Concepts

Completing the Square

Completing the square is an algebraic technique used to rewrite a quadratic expression as a perfect square plus a constant. This method is essential for converting a quadratic equation into vertex form, thus making it easier to identify the vertex and other properties of the parabola.

Graphing a Horizontal Parabola

When the quadratic equation is in the form x = a y² + by + c, the parabola opens either to the right or left. This horizontal orientation means that the roles of x and y are interchanged compared to the more common vertically opening parabolas. Understanding this concept is important for correctly sketching the graph and determining the direction in which the parabola opens.

Parabola

A parabola is a type of conic section defined by a quadratic equation. Its graph is a symmetric curve that represents the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). The vertex is the point where the parabola changes direction, making it a key feature in graphing and analysis.

Vertex Form of a Parabola

The vertex form of a quadratic equation provides a way to immediately read off the vertex of the parabola. By rewriting an equation (whether in x = a y² + by + c form or the standard y = ax² + bx + c form) into vertex form, one can directly identify the turning point of the parabola, which is crucial for sketching its graph.

Transcript

00:01 In problem 12, we have the equation x equals 3y squared plus 6y plus 7.

00:06 And with this, we need to graph this and find the vertex.

00:11 So the first thing we know is that since we have that y -squared term, x -axis is the major axis, opens either to the left or to the right.

00:18 So we're looking to put this in the standard form of x is equal to a y -minus k squared plus h.

00:26 In order to do that, we're going to need to take the right -hand side, and we're going to need to complete the square.

00:32 And that's so we can find that h and k for our vertex, as well as the a, so you can easily identify the behavior of this graph.

00:40 So we have a is equal to, i'm going to put in parentheses this 3y squared plus 6y, that's what we're going to complete the square on, and outside of that we're going to have this constant of plus 7.

00:51 And we're going to have to balance this equation by kind of doing the opposite of whatever we add inside these parentheses.

00:57 We'll need to do the opposite of that outside the parentheses.

01:03 So we need to have a coefficient of 1 on the y.

01:07 So we are going to divide the whole parentheses by 3.

01:11 We're going to get y squared plus 2y.

01:15 Leave that gap there outside the parentheses plus 7.

01:20 So we can take, now complete the square, because we have that coefficient of 1 on the y squared.

01:25 So half of 2, the coefficient of the 1, the y is going to be one, square that, we're just going to have plus one.

01:33 And then we added one inside the parentheses.

01:36 Remember, the coefficient of the parentheses is three.

01:39 So we essentially added three times one to the right -hand side of the equation...

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