In Exercises 27-42, a) Write the equation in the form...

Key Concepts

Completing the Square

This is a technique used to convert a quadratic expression into a perfect square trinomial. By completing the square, one can rewrite a quadratic equation in vertex form, making it easier to identify key features such as the vertex and axis of symmetry.

Vertex Form

Vertex form is a specific way of writing quadratic equations (either y = a(x-h)² + k or x = a(y-k)² + h) that makes the vertex of the parabola readily apparent. It is useful for easily identifying the transformation from a standard parabola and for graphing the equation.

Parabola Orientation

The orientation of a parabola depends on which variable is squared. When the x-term is squared, the parabola opens up or down, while when the y-term is squared, as in the given equation, the parabola opens to the right or left. Understanding the orientation is critical for correctly graphing the curve.

Graphing Parabolas

Graphing involves using the vertex form to identify and plot the vertex along with other significant points such as the focus, directrix, or intercepts. After establishing these key attributes, one can draw the smooth, symmetric curve that characterizes the parabola.

Transcript

00:01 So in this example, we have an equation x equals y squared plus 4y minus 6.

00:08 And we're going to rewrite this in the different form, but we can easily see the vertex, and then we're going to graph it.

00:15 So what we need to do if we need to write it by completing the square is we're going to determine what we need to add to these first two terms to make that a perfect square trinomial.

00:27 So i have x equals y squared plus 4 y plus we're looking for that missing number.

00:36 And the shortcut to find this value is we can cut the middle number in half, that b value, and then we can square it.

00:42 So 4 divided by 2 is 2 and 2 squared is 4.

00:46 So in order to make this a perfect square and complete the square, we need to add 4 to the right hand side.

00:51 Now if we add something to one side, we also need to add it to the opposite side.

00:55 So we end up with x plus 4 equals.

00:59 And now we're going to take this trinomial and we're going to factor it.

01:02 And the beauty of these is that it factors as one binomial squared.

01:06 It's the same binomial twice.

01:08 And in this case, that's y plus 2.

01:10 So it factors as y plus 2 times y plus 2.

01:13 We can write it this way as a shortcut.

01:15 And then lastly, we're going to subtract that 4 back on both sides so that we get the x by itself.

01:21 And we've now fully converted it into our other form of our equation.

01:27 Now if i want to graph it, we usually have our key points that we focus on, which are our vertex and x intercepts and y intercepts.

01:40 So let's see what we can figure out.

01:41 I know from this form i can easily find the vertex now, which is at negative 10 and negative 2.

01:48 This is our x value and our y value.

01:51 Let's go to negative 10.

01:54 Let's build this in here.

01:56 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and down 2...

Please give Ace some feedback