1. 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)
total:

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).
y = a(x + 5)2 + 6 +1
2 = 36a + 6 –6 –6
Substitute the other point.
–4 = 36a
2 = a(1 + 5)2 + 6 +1 36 36 +1 +1 +1 +1 +½ +½

3. 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 +1
(Pythagorean Thm) –5 5 8 +½ –5 +½
+1 labeled graph +1
+1 points plotted +1
total:
+1 right triangle

4. Recall the following algebra tiles:
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. Recall the following algebra tiles:
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. x + 4 x + 4 Therefore, What would be the area of the actual shape?
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. Let’s convert y = x2 + 4x + 9 into vertex form.
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. x + 2 x + 2 y = x2 + 4x + 9 How many “extra” unit tiles are left?
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. PG – 48
How could you complete the square of the quadratic expression and find the vertex for f(x) = x2 + 5x + 2? x
+ 2.5 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. ( )2 Completing the Square Take out the to Write in Vertex Form
worksheet:
Steps:
Example:
1) Write equations in the form _________________.
f(x) = x2 + 6x + 7
y = ax2 + bx + c.
2) _____ the terms with variables together.
Group
f(x) = (x2 + 6x ____) + 7 ____
( )2
= (3)2
= 9
3) Take ____ of “b” and _______ it. _____ and ________ this number to the same side of the equation.
half
square
Add
subtract
f(x) = (x2 + 6x ____) + 7 ____
+ 9
– 9 9 6
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.)
f(x) = (x ____)2 + 7 – 9
+ 3
square binomial
f(x) = (x + 3)2 – 2
5) Simplify the equation.

14. ( )2 ( )2 Practice: = (2)2 = 4 = (5)2 = 25
Rewrite the equations in vertex form using the method for completing the square.
a) f(x) = x2 + 4x + 11
b) f(x) = x2 + 10x
f(x) = (x2 + 4x ____) + 11 ____
+ 4
– 4
f(x) = (x2 + 10x ____) ____
+ 25
– 25
( )2
( )2
= (2)2
= 4
= (5)2
= 25
f(x) = (x ____) – 4
+ 2
f(x) = (x ____)2 – 25
+ 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
f(x) = (x2 + 7x _______) + 2 _______
– 12.25
( )2
= (3.5)2
= 12.25
f(x) = (x ______)2 + 2 – 12.25
+ 3.5
f(x) = (x + 3.5)2 – 10.25

16. Finish today's assignment:
FX ,
& worksheet

17. OLD SLIDES

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

19. x + 2 x + 2 y = x2 + 4x + 9 How many “extra” unit tiles are left?
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) y 2 4 6 8 10 x –2 –4 –6 –8
–10 12 14 16 18
+ 2
(0, 9)
(– 2, 5)
x = –2

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

21. Complete PG - 49

22. PG – 49
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.
a) f(x) = x2 + 6x + 7
b) y = x2 + 4x + 11
y = (x + 3)2 – 2
y = (x + 2)2 + 7
vertex (–3, –2)
vertex (–2, 7)

23. PG – 49
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.
c) f(x) = x2 + 10x
d) y = x2 + 7x + 2
y = (x + 3.5)2 –10.25
y = (x + 5)2 – 25
vertex (–3.5, –10.25)
vertex (–5, –25)