Warm Up for Lesson 3.6

Identify the vertex of each parabola:

(1). y = -(x + 3)2 – 5

(2). y = 2x2 + 7

(3). y = 3(x – 1)2 + 4

(4). y = -5(x + 4)2

(5). Solve: 0 = x2 – 8x – 9

Warm Up Answers for Lesson 3.6

Identify the vertex of each parabola:

(1). y = -(x + 3)2 – 5 V = (-3, -5)

(2). y = 2x2 + 7 V = (0, 7)

(3). y = 3(x – 1)2 + 4 V = (1, 4)

(4). y = -5(x + 4)2 V = (-4, 0)

(5). Solve: 0 = x2 – 8x – 9 x = 9, -1

Graphing Quadratic FunctionsStandard Form and

Completing the Square

Section 3.6

Standard: MM2A3 bc

Essential Question: How do I analyze and graph quadratic functions in standard form?

Let’s begin by squaring a few binomials:

1. (x + 3)2 = _______________

2. (x – 4)2 = _______________

3. (x + 6)2 = _______________

4. (x – 1)2 = _______________

Is there any kind of a pattern that we noticewithin each problem?

Let’s test our theory by factoring a few perfectsquare trinomials:

1. x2 + 10x + 25 = _______________

2. x2 – 6x + 9 = _______________

3. x2 + 14x + 49 = _______________

Finally, let’s look at some examples where we must determine the missing value so that the polynomial is a perfect square trinomial.

1. x2 + 4x + ___

2. x2 – 8x + ___

3. x2 + 12x + ___

Using the information from these few examples,we are going to examine a method called “completing the square”. This method will appear in other areas of mathematics, but in this unit it will be used to change a quadratic from general form to vertex form.

Start with: y = x2 + 6x + 8 and change to

y = (x + 3) 2 -1.

So how does it work?

Example 1: y = x2 + 6x + 8

Rewrite your equation so that there is a space betweenThe 6x and 8, like this:

y = (x2 + 6x + ____ ) + 8

Now we have to think: What value can I add in the space I have so that the first three terms create a perfect square trinomial?

I can determine this quickly and easily by halving themiddle coefficient and then squaring it.

Example 1: y = x2 + 6x + 8

y = (x2 + 6x + ____) + 8

Half of 6 is 3, and 32 = 9, so I will add 9 in the space.

In algebra, we know that we cannot simply add 9 to an equation without doing something to balance it out.

In this case, since we added 9 on the right hand side, let’s also subtract 9 on the right hand side. This is like adding zero to the right side of the equation, which does not change its value.

9 9

Example 1: y = x2 + 6x + 8

y = (x2 + 6x + __9__) + 8 - 9

We’re almost finished! I can factor the polynomial now that it is a perfect square trinomial and I can combine like terms outside of the parentheses. Doing this results in:

y = (x + 3)2 – 1

Now we can easily see the vertex for this quadratic is (-3, -1) and we could graph it more easily.

Example 2: y = x2 + 10x + 7

Example 3: y = x2 - 8x - 7

It can be tricky if our leading coefficient isn’t 1, but we can always make our leading coefficient 1 by factoring.

Example 4: y = -x2 + 6x + 5

Let’s leave a space between the last 2 terms as we have been doing: y = (-x2 + 6x ___) + 5

And let’s factor a -1 out of our polynomial, like this:y = -1(x2 – 6x _____) + 5

y = -1(x2 – 6x _____) + 5

From our previous examples, we know that we need to add 9 inside of the parentheses to complete the square.

But what have we really added?

What? Why?If we distribute the -1 to the term we added (9), we get -9. So since we really added a -9 inside the parentheses, we need to add 9 outside of the parentheses.

Simplifying, this gives us y = -(x – 3)2 + 14

9

9 9

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

(1). Graph: y = x2 – 4x + 8

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

(1). Graph: y = x2 – 4x + 8

x y

Identify each characteristic of: y = x2 – 4x + 8(a). Domain:(b). Range: (c). Vertex:(d). Axis of symmetry:(e). Opens:(f). Max or Min:(g). x-intercept: (h). y-intercept:(i). Extrema: (j). Increasing:(k). Decreasing:(l). Rate of change (-2 ≤ x ≤ 1):

Estimated Rate of change over the interval (-2 ≤ x ≤ 1): y = x2 – 4x + 8

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

(2). Graph: y = x2 – 6x + 5

(2). Graph: y = x2 – 6x + 5

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

(2). Graph: y = x2 – 6x + 5

x y

Identify each characteristic of: y = x2 – 6x + 5 (a). Domain:(b). Range: (c). Vertex:(d). Axis of symmetry:(e). Opens:(f). Max or Min:(g). x-intercept:(h). y-intercept:(i). Extrema:(j). Increasing:(k). Decreasing:(l). Rate of change (-6 ≤ x ≤ -3):