Transformations, Quadratics, and CBRs Jerald Murdock [email protected] .

48
Transformations, Quadratics, and CBRs Jerald Murdock [email protected] www.jmurdock.org

Transcript of Transformations, Quadratics, and CBRs Jerald Murdock [email protected] .

Page 1: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

Transformations, Quadratics, and CBRs

Jerald Murdock

[email protected]

www.jmurdock.org

Page 2: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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

Page 3: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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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Page 4: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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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Page 5: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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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Page 6: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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)

Page 7: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

Step by step transformations get the Job done

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xpipe.sourceforge.net/ Images/RubicsCube.gif

Page 8: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

Create this transformation

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a solution

L3 = - L1 – 3

L4 = - L2 + 4

Page 9: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

“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

Page 10: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

Transformations of Functions

12

3

-1-2

-3

Page 11: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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

Page 12: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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

Page 13: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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)

Page 14: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

Don’t tell students the rules!

Help them discover the rules!

Page 15: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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

Page 16: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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

Page 17: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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

Page 18: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

Ask a kid to walk a parabola

Page 19: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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.

Page 20: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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.

Page 21: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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

Page 22: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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)

Page 23: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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)

Page 24: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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

Page 25: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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)

Page 26: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

A symbolic model

y=ab2 (x−h)2 +k

y=0.340.262 (x−0.9)2 +0.68

Page 27: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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)

Page 28: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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

Page 29: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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

Page 30: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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

Page 31: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

Verification!

-33 = -5(x – 1.5)2 + 12

when x = -1.5 or 4.5

Page 32: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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.

Page 33: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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

Page 34: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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

Page 35: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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

Page 36: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

Verification!

Page 37: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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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Page 38: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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⋅( )

Page 39: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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

Page 40: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

An Important Relationship

ax2 +bx+c→ a(x−h)2 +k

−b2a

=h

c−b2

4a=k

Page 41: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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.

Page 42: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

Parab Program

Page 43: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

Check out the Transform Program

Page 44: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

y = √ (1 – x2) A good function to exhibit horizontal stretches

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.QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

Page 45: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

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

Page 46: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

9

8

7

6

5

4

3

2

1

-1

-2

-4 -2 2 4 6 8 10 12 14

Model with Sketchpad 4

Page 47: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

And if the vertex isn’t available?

• How about finding a, b, and c by using 3 data pairs?

y = ax2 + bx + c

Page 48: Transformations, Quadratics, and CBRs Jerald Murdock Jmurdock@traverse.com .

“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.]