Download - Transformations, Quadratics, and CBRs
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Based on the Discovering Algebra Texts–Key Curriculum Press
• Investigation-based learning
• Appropriate use of technology
• Rich and meaningful mathematics
• Strong connections to the NCTM Content and Process Standards
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Begin by transforming a figure
We’ll plot this figure in L1 and L2 and the transformed image in L3 and L4.
L1 L2
1 1
1 3
1 5
4 5
4 3
1 3
4 1
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-9.4 ≤ x ≤ 9.4
-6.2
≤ y
≤ 6
.2
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Show both original figure and image
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1. slide the figure left 8 units.
2. slide the original figure 6 units down.
3. slide the original figure down 3 units and left 5 units.
L3 = L1 – 8, L4 = L2QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
L3 = L1, L4 = L2 – 6
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L3 = L1 – 5, L4 = L2 – 3
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Show pre-image and image
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4. Reflect figure over the x-axis.
5. Reflect the resulting figure over the y-axis.
6. Shrink the original figure vertically by a factor of 1/2, translate the result 3 units down and left 5 units.
L3 = L1, L4 = - L2QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
L5 = - L1, L6 = - L2
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L3 = L1 – 5, L4 = 0.5 L2 – 3
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An Explanation• To translate a figure to the left 5 units
• New x list = old x list – 5
• To translate a figure to the right 3 units• New x list = old x list + 3
• To slide a figure 6 units down• New y list = old y list – 6
• To reflect a figure over the x-axis• New y list = - (old y list)
• To vertically stretch a figure by factor k• New y list = k(old y list)
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Step by step transformations get the Job done
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xpipe.sourceforge.net/ Images/RubicsCube.gif
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Create this transformation
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a solution
L3 = - L1 – 3
L4 = - L2 + 4
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“I’m very well acquainted, too, with matters mathematical.I understand equations both the simple and quadratical.About binomial theorem I am teeming with a lot of news. . .With many cheerful facts about the square of the hypotenuse!”
The Major General’s Song from The Pirates of Penzance by Gilbert and Sullivan
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Transformations of Functions
12
3
-1-2
-3
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Translations of y = x2
1 2 3–1–2–3
1
2
3
4
5
–1 1 2 3–1–2–3
1
2
3
4
5
–1 1 2 3–1–2–3
1
2
3
4
5
–1
1 2 3–1–2–3
1
2
3
4
5
–1
y=x2 +2
y= x−2( )2 y= x+1( )2y= x−2( )2 +1
1 2 3–1–2–3
1
2
3
4
5
–1
y=x2
1 2 3–1–2–3
1
2
3
4
5
–1
y=x2 −1
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Original equation y = x2
• To move a function 3 units right we must replace the old x in the original equation with x – 3.
y = (x – 3)2
• Similarly to move a function 2 units left we must replace the old x in the equation with x + 2.
y = (x + 2 )2
• To move a function to the right 3 units new x = old x + 3
or old x = new x – 3
• And to move a function 2 units down we must replace the old y in the equation with y + 2. y + 2 = x2 or y = x2 – 2
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Make my graph
y = -2(x + 3)2 + 5 y = -(x – 1)2 + 3
y = 0.5(x – 3)2 y = -(x + 2)2 + 4
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(-3,5)
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(1,3)
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(3,0)
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(-2,4)
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Don’t tell students the rules!
Help them discover the rules!
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Write my equation
Use the point (-2, 3) to determine the value of a
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(-3,5)This equation sets the vertex at (-3, 5)
y = a(x + 3)2 + 5
3 = a(-2 + 3)2 + 5
y = –2(x + 3)2 + 5
3 = a(1)2 + 5
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Program:CBRSET
• Prompt S,N
• round (S/N, 5)I
• If I > 0.2:-0.25 int(-4I) I
• Send ({0})
• Send ({1, 11, 2, 0, 0, 0})
• Send({3, I, N, 1, 0, 0, 0, 0, 1, 1})
CBR will collect 40 data pairs in 6 seconds
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Program:CBRGET
• Send ({5,1})
• Get (L2)
• Get (L1)
• Plot 1 (Scatter, L1, L2, · )
• ZoomStat
Program will transfer data from CBR, set window, and plot graph on your calculator
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Ask a kid to walk a parabola
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A single ball bounce• Set CBR to collect 20 data points during one second.• Drop a ball from a height of about 0.5 meters and trigger
the CBR from about 0.5 meters above the ball. Transfer data to all calculators.
• Model your parabola with an equation of the form
y = a(x – h)2 + k
Find a by trial & error or find a by substituting another choice of x and y.
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Modeling provides students with
• a logical reason for learning algebra- kids don’t
ask “When am I ever going to use this?”
• Students can gain important conceptual understandings and build up their “bank” of basic symbolic algebra skills.
• an integration of algebra with geometry, statistics, data analysis, functions, probability, and trigonometry.
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identifies the location of the vertex after a translation from (0, 0) to (h, k). This parabola shows a vertical scale factor, a, and horizontal scale factor, b.
In General
y−ka
=x−h
b⎛ ⎝
⎞ ⎠
2
or y=ax −h
b⎛ ⎝
⎞ ⎠
2
+k
y=ab2 (x−h)2 +k
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Locate your vertex (h, k)• Then find your stretch factors (a & b) by selecting
another data point (x1,y1).
y−ka
⎛ ⎝ ⎜
⎞ ⎠ ⎟ =
x−hb
⎛ ⎝ ⎜
⎞ ⎠ ⎟
2
(h, k) = (0.86, 0.6)
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A transformation of y = x2
y−0.6a
⎛ ⎝ ⎜
⎞ ⎠ ⎟ =
x−0.86b
⎛ ⎝ ⎜
⎞ ⎠ ⎟
2
a=y1 −k =0.18−0.6=−0.42
b=x1 −h=1.14−0.86=0.28
Choose point (x1,y1) = (1.14, 0.18)
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A transformation of y = x2
y−ka
⎛ ⎝ ⎜
⎞ ⎠ ⎟ =
x−hb
⎛ ⎝ ⎜
⎞ ⎠ ⎟
2
y−0.6−0.42
⎛ ⎝ ⎜
⎞ ⎠ ⎟ =
x−0.860.28
⎛ ⎝ ⎜
⎞ ⎠ ⎟
2
y=−0.42x −0.86
0.28
⎛ ⎝ ⎜
⎞ ⎠ ⎟
2
+0.6
y = –5.36(x – 0.86)2 + 0.6
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Another bounce exampley−k
a
⎛ ⎝ ⎜
⎞ ⎠ ⎟ =
x−hb
⎛ ⎝ ⎜
⎞ ⎠ ⎟
2
Choose (x1,y1)= (1.16,1.02)
a=y1 −k =1.02−0.68=0.32
b=x1 −h=1.16−0.9=0.26
(h, k) = (0.9, 0.68)(0.9, 0.68)
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A symbolic model
y=ab2 (x−h)2 +k
y=0.340.262 (x−0.9)2 +0.68
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A Case for Vertex Form
• y = a(x – h)2 + k– Vertex is visible at (h, k)– Easy to approximate the value of a– Supports the order of operations– Easy to solve for x (by undoing)
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Evaluate by emphasizing the order of operations
• Evaluate this expression when x = 4.5
3
9
- 45
- 33
4.5• Subtract 1.5
• Square
• Multiply by –5
• Add 12
•So, the value of the expression at x = 4.5 is –33
−5(x −1.5)2 +12
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An organization template
– (1.5)
( )2
x (–5)
+ (12)
Operations
Pick x = 4.5
Results
−5(x −1.5)2 +12
4.5
3
9
–45
–33
Evaluate this expression when x = 4.5
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Solve equations by undoing
x =
– (1.5)
( )2
x (-5)
+ (12) – (12)
÷ (-5)
±
+ (1.5)
-33
-45
4.5, -1.5Operations
Pick x
Undo Results
± 3
9
−5(x −1.5)2 +12=−33
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Verification!
-33 = -5(x – 1.5)2 + 12
when x = -1.5 or 4.5
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What question is asked?
x =
– (0.9)
( )2
x (5.03)
+ (0.68) – (0.68)
÷ (5.03)
±
+ (0.9)
1
0.32
1.15…, 0.647…
Operations
Pick x
Undo Result
± 0.252…
0.0636…
5.03(x – 0.9)2 + 0.68 = 1QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
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The power of visualization and symbolic algebra
3 4
3
4
7
79 12
12 16
72=(3 + 4)2
9 + 12 + 12 + 16 = 49
x 4
x
4
x + 4
x + 4x2 4x
4x 16
(x + 4)2 = x2 + 4x + 4x + 16
(x + 4)2 = x2 + 8x + 16
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x
x x2
Polynomial Vertex Form
• Use a rectangular model to help complete the square. x
2 +14x
+ 7
→ x+7( )2−49
7x
49
−49
7x + 7
→ x+7( )2−49
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x
x x2
Polynomial Vertex Form
• Use a rectangular model to help complete the square. x2 +12x ( )+13
6x
+ 6
→ x+6( )2 −36+136x
+ 6 36
−36
x+6( )2 −23
+13
→
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Verification!
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What question is asked?
x =
+ (6)
( )2
– (23) + (23)
±
– (6)
0
± (23) – 6
Operations
Pick x
Undo Result
± (23)
23
x2 + 12x + 13 = 0
Or (x + 6)2 – 23 = 0
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x
x x2
General Vertex Form
• Use a rectangular model to help complete the square.
2x2 −8x +9
−2x
− 2
→ 2 x−2( )2 −8+9−2x
−2 4
−8
2 x−2( )2 +1
+9
→ 2 x2 −4x( )+9
2⋅( )
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x
x x2
General form and the vertex
b2a
x
b2a
→ a x +b2a
⎛ ⎝ ⎜
⎞ ⎠ ⎟
2
+c−b2
4a
b2a
x
b2a
b2
4a2
+c−ab2
4a2 a⋅( )ax2 +bx+c =a(x2 +
ba
x )+c
−b2a
,c−b2
4a
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ Vertex is
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An Important Relationship
ax2 +bx+c→ a(x−h)2 +k
−b2a
=h
c−b2
4a=k
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Rolling Along
• Place the CBR at the high end of the table.
• Roll the can up from the low end. It should get no closer than 0.5 meters to the CBR.
• Collect data for about 6 seconds.
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Parab Program
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Check out the Transform Program
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y = √ (1 – x2) A good function to exhibit horizontal stretches
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QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
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5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
-0.5
1 2 3 4 5 6 7 8
B: (3.53, 2.84)
f x( ) = -0.15 ⋅ -3.53x( )2+2.84
B
A
Paste a picture into Sketchpad
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9
8
7
6
5
4
3
2
1
-1
-2
-4 -2 2 4 6 8 10 12 14
Model with Sketchpad 4
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And if the vertex isn’t available?
• How about finding a, b, and c by using 3 data pairs?
y = ax2 + bx + c
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“If we introduce symbolism before the concept is understood, we force students to memorize empty symbols and operations on those symbols rather than internalize the concept.”
[Stephen Willoughby, Mathematics Teaching in the Middle School, February 1997, p. 218.]