Complete the square of each quadratic expression. Then graph each function using graphing techni

Name: Problem 72, Chapter 3: Algebra and Trigonometry
Uploaded: 2021-06-22T17:38:33-08:00
Duration: 4 min 38 s
Channel: Sarah Lewites
Description: Complete the square of each quadratic expression. Then graph each function using graphing techniques. f(x)=-2 x^2-12 x-13

step 1 So in this problem, we have two pieces. First, we're going to complete the square on this quadra

Complete the square of each quadratic expression. Then graph...

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two that can be expressed in the form ax² + bx + c. They are fundamental in algebra and appear in various contexts such as physics, engineering, and economics. The standard form is useful for algebraic manipulation, while different forms like vertex form (a(x-h)² + k) provide insights into the graph's properties such as its vertex, axis of symmetry, and direction of opening.

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in a form that reveals important features of the quadratic function. This involves rearranging and factoring the expression so that one part of it becomes a perfect square trinomial. This process transforms the standard quadratic form into the vertex form, making it easier to analyze and graph the function.

Vertex Form

The vertex form of a quadratic function is given by a(x-h)² + k, where (h, k) is the vertex of the parabola. This form makes it straightforward to identify the vertex and determine the axis of symmetry (x = h). It also facilitates the analysis of transformations, such as horizontal and vertical shifts, stretching, and reflections, thereby providing a clear geometric interpretation.

Graphing Techniques

Graphing quadratic functions involves using the vertex form and other key properties such as the axis of symmetry, direction (upward or downward opening), and the y-intercept. By identifying the vertex and plotting additional points, one can sketch an accurate graph of the parabola. Understanding transformations such as translations, reflections, and dilations further aids in creating a precise graphical representation of the quadratic function.

Transcript

00:01 So in this problem, we have two pieces.

00:03 First, we're going to complete the square on this quadratic.

00:07 And then after we've completed the square, we're going to use graphing techniques to sketch a graph.

00:12 So we'll begin by completing the square.

00:14 Completing the square involves transforming the function c value so that we have a perfect square trinomial.

00:20 Currently, this function has a c value of minus 13, which is not giving us a perfect square trinomial.

00:27 So to begin, we want to add the 13 to the other side so that we can, focus on getting a perfect square trinomial on the right side.

00:37 We haven't changed the value of the function.

00:39 We've just rearranged it so that it's more clear to see what we need to do.

00:43 Our second step will involve factoring the right side.

00:48 We can factor out a negative 2 from both of these terms.

00:53 So we get x squared plus 6x.

00:56 Again this is in the process of creating a perfect square trinomial.

01:01 Now to find what we need for a perfect square ternomial, we need a a c value here, a new c value, that gets this function or this trinomial to be a perfect square.

01:13 Now to determine what c value we need, we have a formula.

01:17 We use b over two squared.

01:21 In this case, our b value is six.

01:23 So we're doing six over two squared or three squared, which is nine.

01:29 So we need this c value to be nine, and then we will have a perfect square trinomial.

01:35 So we're going to add nine to this side.

01:37 But we can't just go around adding nine to things.

01:39 We have to also make sure we add the same amount to the other side.

01:43 This one is a little bit interesting, though, because we're not just adding a plus nine.

01:48 We're adding negative two copies of the plus nine, which means we're actually subtracting 18 from this side.

01:55 So to stay totally even, we also need to subtract 18 from the other side.

02:01 Then we'll be fair and balanced.