Rewrite each equation in the form x=a(y-k)^2+h by completing the square and graph it. x=-4 y^2-8

Rewrite each equation in the form x=a(y-k)^2+h by completing...

Key Concepts

Completing the Square

Completing the square is a method used to rewrite a quadratic expression as a perfect square plus or minus a constant. This technique simplifies the process of converting a quadratic equation from standard form to vertex form, thereby making it easier to analyze and graph.

Vertex Form of a Parabola

The vertex form of a quadratic equation, often written as x = a(y - k)^2 + h or y = a(x - h)^2 + k, clearly indicates the vertex of the parabola and its orientation. This form is useful for identifying the maximum or minimum point of the parabola, as well as understanding its shape and direction.

Graphing Quadratic Functions

Graphing quadratic functions involves plotting the vertex, determining the axis of symmetry, and identifying the direction the parabola opens based on the coefficient 'a'. Using the vertex form makes it easier to accurately sketch the parabola by providing key information about its transformation from the parent function.

Transcript

00:01 So in this example, i'm given an equation of our parabola that opens sideways.

00:07 Then we're going to convert this equation to our other form.

00:11 So we can easily see the vertex, and we're going to use those data points to graph it.

00:15 So the first thing when we're converting is we're going to complete the square, but we can't do that really unless the y squared has a 1 as a coefficient.

00:22 So i need to divide this negative 4 so that we can get that 1 y squared we're looking for.

00:28 We always have to do things on both sides.

00:30 And in the end, we'll end up multiplying that 4 back.

00:33 So don't worry about any fractions or other issues for now.

00:37 So this will become a y squared, negative 8 divided by negative 4 is 2, and 10 divided by 4.

00:46 We can actually just leave it like that for now.

00:52 Okay.

00:53 So from here, we're going to group these first two terms.

00:56 We're going to figure out what to add to make that a perfect square.

00:59 So our shortcut to determine that is we can take this middle, b value, and divide it in half.

01:05 So 2 divided by 2 is 1, and then we square that value.

01:09 So 1 squared is 1.

01:11 So that means if i add 1, i'm going to end up with a perfect square, so i'm going to go ahead and do that.

01:16 But if i add one on the same side, i also need to subtract one to make sure that they zero out, and i'm not really changing my equation.

01:23 Now what this lets me do is factor this trinomial.

01:26 So i'm going to keep the x over negative 4.

01:29 This trinomial is going to factor is y plus 1 squared, plus we have 10 over 4 minus...

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