Assignment, pencil red pen, highlighter, textbook, GP notebook, calculator, cut–out algebra tiles

1. total: Assignment, pencil red pen, highlighter, textbook, GP notebook, calculator, cut–out algebra tiles U4D4 Have out: Bellwork: 1. Write the equation for a parabola that has a vertex of (–5, 6) and goes through the point (1, 2). 2. Given the points (–4, 3) and (4, –1), compute: a) the slope b) the distance (hint: graph the points, make a right triangle, find the length of the hypotenuse)

2. Bellwork: 1. Write the equation for a parabola that has a vertex of (–5, 6) and goes through the point (1, 2). Solve for a. y = a(x – h)2 + k +1 2 = a(6)2 + 6 Substitute the vertex (–5, 6). 2 = 36a + 6 y = a(x + 5)2 + 6 +1 –6 –6 Substitute the other point. –4= 36a 2 = a(1 + 5)2 + 6 +1 36 36 +1 +1 +1 +1 +½ +½

3. total: Bellwork: 2. Given the points (–4, 3) and (4, –1), compute: a) the slope b) the distance y +1 5 +1 +½ +½ 4 x (Pythagorean Thm) +1 –5 5 8 +½ –5 +½ +1 labeled graph +1 +1 points plotted +1 +1 right triangle

4. PG – 46 Recall the following algebra tiles: 1 1 1 x x2 x x These are useful for finding the area of squares and rectangles. x 1

5. PG – 46 Recall the following algebra tiles: 1 1 1 x x2 x x These are useful for finding the area of squares and rectangles. x 1 Algebraically, we can represent the following as: x2 + 8x + 10

6. PG – 46 Today we are going to take equations in standard form and convert them to vertex form. That is, we are going to take y = x2 + 8x + 10 and convert it to the form y = a(x – h)2 + k By changing equations to vertex form, we can quickly graph the parabolas.

7. PG – 46 Let’s take the tiles that represent x2 + 8x + 10 and make as complete a square as possible. x + 4 If the square was completed, what would be the length of each side of the square? x x + 4 What would be the area of the complete square? (x + 4)2 However, we don’t have a complete square. How many “unit” squares do we need to subtract to get the actual area? + 4 We need to subtract 6 unit tiles.

8. PG – 46 Therefore, What would be the area of the actual shape? x + 4 (x + 4)2 – 6 x What can we conclude about x2 + 8x + 10 and our answer above? x2 + 8x + 10 = (x + 4)2 – 6 Therefore, y = x2 + 8x + 10 can be written as y = (x + 4)2 – 6 + 4 What is the vertex of the parabola y = (x + 4)2 – 6? vertex (–4, –6)

9. PG – 47 Let’s convert y = x2 + 4x + 9 into vertex form. The equation can be represented by the following tiles. y = x2 + 4x + 9

10. PG – 47 y = x2 + 4x + 9 How many “extra” unit tiles are left? There are 5 extra tiles left over. x + 2 a) What new equation can we write? y = (x + 2)2 + 5 x b) Name the vertex vertex (–2, 5) + 2

11. How could you complete the square of the quadratic expression and find the vertex for f(x) = x2 + 5x + 2? PG – 48 + 2.5 x x What about the extra bar? Use force! Split the bar in half. + 2.5 a) How many unit squares, including parts of squares are we missing? 1 + 1 + ½ + ½ + ½ + ½ + ¼ = 4.25 b) What expression represents the complete square? c) Write the vertex form of the equation. (x + 2.5)2 – 4.25 (x + 2.5)2 y =

12. Is there a faster way to complete the square without having to use algebra tiles?

13. 9 6 Completing the Square to Write in Vertex Form Take out the worksheet: Example: Steps: 1) Write equations in the form _________________. f(x) = x2 + 6x + 7 y = ax2 + bx + c. Group 2) _____ the terms with variables together. f(x) = (x2 + 6x ____) + 7 ____ ( )2 = 9 = (3)2 half square 3) Take ____ of “b” and _______ it. _____ and ________ this number to the same side of the equation. Add subtract – 9 + 9 f(x) = (x2 + 6x ____) + 7 ____ 9 and –9 cancel each other out, so we are not changing the equation, just rewriting it. 3 3 perfect square trinomial 4) Rewrite the ___________________ (x2+ 6x + 9) as a ______________. (Hint: take half of “b” or just factor.) + 3 f(x) = (x ____)2 + 7 – 9 square binomial f(x) = (x + 3)2 – 2 5) Simplify the equation.

14. Practice: Rewrite the equations in vertex form using the method for completing the square. a) f(x) = x2 + 4x + 11 b) f(x) = x2 + 10x – 4 + 4 – 25 + 25 f(x) = (x2 + 4x ____) + 11 ____ f(x) = (x2 + 10x ____) ____ ( )2 ( )2 = 4 = (2)2 = 25 = (5)2 f(x) = (x ____)2 + 11 – 4 f(x) = (x ____)2 – 25 + 2 + 5 f(x) = (x + 2)2 + 7 f(x) = (x + 5)2 – 25

15. Practice: Rewrite the equations in vertex form using the method for completing the square. c) f(x) = x2 + 7x + 2 – 12.25 + 12.25 f(x) = (x2 + 7x _______) + 2 _______ ( )2 = 12.25 = (3.5)2 f(x) = (x ______)2 + 2 – 12.25 + 3.5 f(x) = (x + 3.5)2 – 10.25

16. Finish today's assignment: FX 46 - 48, 52 - 54 & worksheet

17. OLD SLIDES

18. y 10 8 6 4 2 x –2 2 4 6 8 10 –10 –8 –6 –4 –2 –4 –6 –8 –10 PG – 46 b) Sketch the graph of y = (x + 4)2 – 6. x = – 4 y = (x + 4)2 – 6 (0, 10) (–4, –6)

19. y 18 16 14 12 10 8 6 4 2 x –4 –2 2 4 6 8 10 –8 –6 –10 –2 PG – 47 y = x2 + 4x + 9 How many “extra” unit tiles are left? There are 5 extra tiles left over. x + 2 a) What new equation can we write? y = (x + 2)2 + 5 x b) Name the vertex, and sketch the graph. vertex (–2, 5) + 2 (0, 9) (– 2, 5) x = –2

20. y 10 8 6 4 2 x –2 2 4 6 8 10 –10 –8 –6 –4 –2 –4 –6 –8 –10 PG – 48 c) Write the vertex form of the equation. + 2.5 x y = (x + 2.5)2 – 4.25 d) Name the vertex and sketch the graph. x + 2.5 (–2.5, –4.25) x = –2.5

21. Complete PG - 49

22. For each quadratic function use the idea of completing the square to write it in the vertex form. Then state the vertex of each parabola. PG – 49 b) y = x2 + 4x + 11 a) f(x) = x2 + 6x + 7 y = (x + 3)2 – 2 y = (x + 2)2 + 7 vertex (–3, –2) vertex (–2, 7)

23. For each quadratic function use the idea of completing the square to write it in the vertex form. Then state the vertex of each parabola. PG – 49 d) y = x2 + 7x + 2 c) f(x) = x2 + 10x y = (x + 3.5)2 –10.25 y = (x + 5)2 – 25 vertex (–3.5, –10.25) vertex (–5, –25)