Transformations, Quadratics, and CBRs Jerald Murdock

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

Begin by transforming a figure We’ll plot this figure in L 1 and L 2 and the transformed image in L 3 and L 4. L1L L1L ≤ x ≤ ≤ y ≤ 6.2

Show both original figure and image 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. L 3 = L 1 – 8, L 4 = L 2 L 3 = L 1, L 4 = L 2 – 6 L 3 = L 1 – 5, L 4 = L 2 – 3

Show pre-image and image 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. L 3 = L 1, L 4 = - L 2 L 5 = - L 1, L 6 = - L 2 L 3 = L 1 – 5, L 4 = 0.5 L 2 – 3

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)

Step by step transformations get the Job done xpipe.sourceforge.net/ Images/RubicsCube.gif

Create this transformation a solution L 3 = - L 1 – 3 L 4 = - L 2 + 4

“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

Transformations of Functions

Translations of y = x 2

Original equation y = x 2 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 = x 2 or y = x 2 – 2

Make my graph y = -2(x + 3) y = -(x – 1) y = 0.5(x – 3) 2 y = -(x + 2) (-3,5) (1,3) (3,0) (-2,4)

Don’t tell students the rules! Help them discover the rules!

Write my equation Use the point (-2, 3) to determine the value of a (-3,5) This equation sets the vertex at (-3, 5) y = a(x + 3) = a(-2 + 3) y = –2(x + 3) = a(1) 2 + 5

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

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

Ask a kid to walk a parabola

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.

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.

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

Locate your vertex (h, k) Then find your stretch factors (a & b) by selecting another data point (x 1,y 1 ). (h, k) = (0.86, 0.6)

A transformation of y = x 2 Choose point (x 1,y 1 ) = (1.14, 0.18)

A transformation of y = x 2 y = –5.36(x – 0.86)

Another bounce example Choose (x 1,y 1 )= (1.16,1.02) (h, k) = (0.9, 0.68) (0.9, 0.68)

A symbolic model

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)

Evaluate by emphasizing the order of operations Evaluate this expression when x = Subtract 1.5 Square Multiply by –5 Add 12 So, the value of the expression at x = 4.5 is –33

An organization template – (1.5) ( ) 2 x (–5) + (12) Operations Pick x = 4.5 Results –45 –33 Evaluate this expression when x = 4.5

Solve equations by undoing x = – (1.5) ( ) 2 x (-5) + (12)– (12) ÷ (-5) ± ±  + (1.5) , -1.5 Operations Pick x Undo Results ± 3 9

Verification! -33 = -5(x – 1.5) when x = -1.5 or 4.5

What question is asked? x = – (0.9) ( ) 2 x (5.03) + (0.68)– (0.68) ÷ (5.03) ± ±  + (0.9) …, 0.647… Operations Pick x UndoResult ± 0.252… … 5.03(x – 0.9) = 1

The power of visualization and symbolic algebra

Polynomial  Vertex Form Use a rectangular model to help complete the square.

Polynomial  Vertex Form Use a rectangular model to help complete the square.

Verification!

What question is asked? x = + (6) ( ) 2 – (23)+ (23) ± ±  – (6) 0 ±  (23) – 6 Operations Pick x UndoResult ±  (23) 23 x x + 13 = 0 Or (x + 6) 2 – 23 = 0

General  Vertex Form Use a rectangular model to help complete the square.

General form and the vertex Vertex is

An Important Relationship

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.

Parab Program

Check out the Transform Program

y = √ (1 – x 2 ) A good function to exhibit horizontal stretches

Paste a picture into Sketchpad

Model with Sketchpad 4

And if the vertex isn’t available? How about finding a, b, and c by using 3 data pairs? y = ax 2 + bx + c

“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.]