Today:

Review last Khan Academy Topic Review: Graphing, Solving by taking the square root

New Solving Method: Completing the Square Class Work 4.2 front & back

Name five important parts of a parabola

1. Axis of Symmetry2. Vertex3. y-intercept4. y-intercept translated5. Solution(s)

How do we find each of these?

Axis of symmetry: Use the formula: -b2aVertex:

The AOS is the x-coordinate of the vertex. To find the y-coordinate, plug the value of x into the equation and solve for y.y-intercept

Write a quadratic equation in standard form: ax2 + bx + c = 0

The y-intercept is the value of 'c' in the quadratic equation.

y-intercept translated

Determine the distance of the y axis from the AOS. Then, count the same distance on the other sideof the AOS. The y value will be the same.

The solutions are where the graph crosses the x-intercept. Right now, our method for finding solutions is by factoring.

5. Solution(s)

5. Solution(s)

The solutions are where the graph crosses the x-intercept. Right now, our method for finding solutions is by factoring.

We will be learning 3 more methods of finding solutions:

1. By using square roots.2. By completing the square3. by using the quadratic function

Quadratic Equations:

Class Work: 5 AssignmentsKhan Academy Topics: 9Tests: 2

1st Test: a. Graphing Quadratic Equationsb. Solving Equations by factoring/Taking the square

root2nd Test:

a. Solving Quadratic Equations by Completing the Square/ Using the quadratic function

Graphing Quadratic Functions

In order, list the steps needed to solve the following equation

Graphing Quadratic Functions1. Parabola’s which have one solution (actually two identical solutions), are always the graphic form of what type of equation.

2. Complete the factored form of this PST (x )2

3. Finally, write the original equation for the parabola shown.

Perfect Square Trinomial

-2

x2 - 4x + 4

Warm-Up/Review:

Graphing Quadratic FunctionsWarm-Up/Review:

Find the solution(s) to: 4x2 + 12x + 9

Graphing Quadratic FunctionsWarm-Up/Review:

A graph of a quadratic function has x intercepts of (-5,0) and (3,0). What is the axis of symmetry?

The half-way point between two solutions is always the axis of symmetry.

Find the vertex and four other points, then draw the parabola.

Graphing Quadratic Functions

3x2 - 96 = 0

x = √32 =....

Completing the Square

Completing the SquareFactoring “unfactorable” 2nd degree trinomials

• We have learned earlier that a perfect square trinomial can always be factored.

• Therefore, if we have a trinomial we cannot factor using integers, we can change it in such a way that we are dealing with a perfect square trinomial.

Completing the Square:

• Recall that a perfect square trinomial is always in the form:

• Therefore, we have to change the polynomial so that it fits the form.

• To really learn this, go through each step of the process. Your goal should be to learn the steps in order.

22 2 baba

Completing the Square:

Step 1 Divide by the leading coefficient to set the a-value to 1.

2

2

2 16 20 0

2

8 10 0

x x

x x

Completing the Square:

Step 2 Re-write the equation in the form ax + by = c

2

2

2

8 10 0

8 10 10 10

18 0

0

x x

x x

x x

Completing the Square:

The equation we are going to solve is the following…

By testing whether or not the factors of c can sum to equal b, we can determine if the trinomial is factorable. This trinomial is not factorable in its present form. However, with our new tool, we can solve this previously 'unsolvable' quadratic.

2

2 16 20 0x x

There are five steps in this process, let's write them down.

Completing the Square:

Step 3Find one-half of the b value.

Add the square of that number to both sides.

2

2

2

2 2

8 10

8 104 4

16 28 6

x x

x x

x x

2

2

b

Completing the Square:

Step 4A) Re-write the perfect square trinomial as a binomial squared.

B) Find the square root of each side of the equation.

2

24

8 16 26

26

264

x

x

x x

Completing the Square:

Step 5 Solve for x.

4 2

4

2

6

264

6

4

4

x

x

x

Completing the Square:

Example 2

x2 - 16x +15 = 0Re-write the equation in the form ax + by = c

Divide by leading coefficient

x2 - 16x = -152

2

b

Take one-half of b, then square it. Add the square to both sides.

x2 - 16x + 64 = -15 + 64

Simplify both sides.x2 - 16x + 64 = -15 + 64

(x - 8) 2 = 49; then what?

Completing the Square:

Completing the Square:

1) Find the value of

2

2

b

2x bx

2. Add the value to the expression, this completes the square

2 6x x 2 10x x

Completing the Square:

Graphing Quadratic FunctionsClass Work 4.2