Project AMP Dr. Antonio Quesada Director, Project AMP

Project AMP Dr. Antonio Quesada – Director, Project AMP

LEARNING LAB 2 EXPLORATIONS on y = a(x + h)n + k Brian Reeves & Karen Kapes (Cuyahoga Falls) PROJECT AKRON WEB (Created - June 22, 2000)

SUBJECT Advanced Algebra GRADE 11-12 STRAND ALGEBRA II

TOPIC Families of Functions of the forms y= x , y=x2, y=x3 OBJECTIVES Students will plot and graph parent functions and translations and stretching of parent functions

Students will explore by using a graphing calculator translating along the x-axis and the y-axis stretching by changing the lead coefficient

Students will generalize observations about shifts, discovering patterns in their graphs

MATERIALS Worksheet (at end of this lab), graph paper and straight edge; graphing calculators

EXPECTED TIME 3-5 days Project AMP Dr. Antonio Quesada – Director, Project AMP

LEARNING LAB 2 EXPLORATIONS on y = a(x + h)n + k Brian Reeves & Karen Kapes (Cuyahoga Falls) PROJECT AKRON WEB (Created June 22, 2000)

TOPIC Families of equations - linear, absolute value, quadratic, cubic

CONCEPT Families of functions (y=x, y= x , y=x2, y=x3) can be characterized and distinguished according to/by common characteristics. (OPO, Mathematics: Gr 9 - #12; Gr 12 - #5, 6, 10, 11 , 13) (Ohio HSGQE #4, 5, 6, 7, 12, 15, 16) (Summit County Academic Standards: Mathematics: Gr. 11 - #10, 11, 18, 19, 20)

LEARNING Students will plot and graph parent functions and translations OBJECTIVES and stretching of parent functions.

Students will explore by using a graphing calculator translating along the x-axis and the y-axis stretching by changing the lead coefficient

Students will generalize observations about shifts, discovering patterns in their graphs

ASSUMED It is assumed that students doing this lab SKILLS have a working knowledge of terms, such as, domain, plotting points have a basic knowledge of absolute value, quadratic, and cubic functions have graph paper, straight edges, and graphing calculators available

PROJECT TASKS A) Accurately sketch on graph paper - using x-y tables over the domain -10  x  10 the graphs of the parent functions y = x y= x y = x2 y = x3

B) Confirm the graphs from Part A using a graphing calculator by 1) verifying the tables 2) verifying the plot Project AMP Dr. Antonio Quesada – Director, Project AMP

C Plot the parent graph y = x on t graphing calculator. (Remember this is y = 1x, where 1 is the slope.) 1) *Substitute 0.5 for the 1 and compare the 2 graphs. (So y= 0.5x or y = x is a child of the parent y = x.) Compare 2) Repeat substituting 0.8, 0.3, 0.1 for the 1in the parent function (y = 1x). Record observations on the table provided. 3) Repeat substituting 2, 3, 5 for the 1 in the parent function (y = 1x). Record observations on the table provided. 4) Repeat substituting -0.3, -0.5, -0.8, -2, -3 for the 1 in the parent function (y = 1x). Record observations on the table provided. 5) Keep the parent graph and delete any other graphs that are stored. This time add the new number 2 to the x (Parent function is y = x or y =x+0; child becomes y = x+2. Compare the parent and the child graphs. Repeat, substituting 3 his time. 6) Keep the parent graph and add the new number  after the x term (Parent function is y = x or y = x+0.) Compare the parent and the child graphs. Repeat, substituting 5 this time and compare again. 7) Keep the parent graph and add the new number - after the x term (Parent function is y = x or y = x+0.) Compare the parent and the child graphs. Repeat, substituting 5 this time and compare again. 8) Keep the parent graph and add the new number  after the x term (Parent function is y = x or y = x+0.) Compare the parent and the child graphs. Repeat, substituting 4 this time and compare again. 9) Keep the parent graph and add the new number - after the x term Parent function is y = x or ythis time and compare again. 10) Record your conclusions on the table.*

D) Plot the parent graph y = x on the graphing calculator. (Remember this is y = 1x. Repeat between the * in Part C for this function. For Parts 5, 6, and 7 consider the parent as y =x+0 and for Parts 8, and 9 consider the parent as y = x +0.

E) Plot the parent graph y = x2 on the graphing calculator. (Remember this is y = (1x)2.) Repeat between the * in Part C for this function. For Parts 5, 6 and 7 consider the parent as y = (x+0)2 and for Parts 8 and 9 consider the parent as y = x2+0. Project AMP Dr. Antonio Quesada – Director, Project AMP

F) Plot the parent graph y = x3 on the graphing calculator. (Remember this is y = (1 x)3. . Repeat between the * in Part C for this function. For Parts 5, 6, and 7 consider the parent as y = (x+0)3 and for Parts 8 and 9 consider the parent as y = x3+0.

G) Review your work and list any similarities and differences that you observe s you look at all of your work.

H) Develop 2 examples using the general function y = a(x + h)n + k for different values of a, h, k, and n ( n = 1, 2, 3). Graph the function on the graphing calculator and state how the parent function was transformed and/or stretched. Find any zeros shown on your graph. (TRACE can help.) Prepare to present each example to the class, including graphs and tables.

LEARNING This lesson is designed to be a 3-5 day or double block STRATEGIES self-contained lesson with a worksheet.

The worksheet (included at the end of this lab) is used by students in recording their observations. Students explore and manipulate function families in their groups ot 2 or 3 while investigating translations and stretches. The lab concludes with each group presenting functions that are a composite of the types investigated in the lab.

Students might require instruction / review in using the graphing calculator.

Student grouping can be according to pre-assigned cooperative learning groups or can be either assigned or student selected for this assignment.

The teacher prompts and/or assists groups as requested and/or as needed.

CLASSROOM / Cooperative group work followed by group presentations INFORMATION and class discussion and generalization. MANAGEMENT

ASSESSMENT Observation of student work, time, and focus on task

Cooperation within the groups

Group presentations of function families Project AMP Dr. Antonio Quesada – Director, Project AMP

Student observations and notes from discussion Project AMP Dr. Antonio Quesada – Director, Project AMP

TOOLS & Worksheet; graph paper and straight edge; graphing calculator RESOURCES

RESULT Among members of each group in performing tasks SHARING Between groups in sharing observations

Whole class summary and generalizations regarding slope-intercept form Project AMP Dr. Antonio Quesada – Director, Project AMP

WORKSHEET Name ______EXPLORATIONS ON y = a(x+h)n + k ______

For this project, your group will be looking at a variety of functions and their graphs. As you work together with the help of graph paper and graphing calculators, you will make observtions about what happens to the parent graph of the set. Record your observations on the attached worksheet. Then complete the project by working with your own functions according to the guidelines given.

A) Accurately sketch on graph paper - using x-y tables over the domain -10  x  10 - the graphs of the parent functions y = x y= x y = x2 y = x3

B) Confirm the graphs from Part A using a graphing calculator by 1) verifying the tables 2) verifying the plot

C Plot the parent graph y = x on t graphing calculator. (Remember this is y = 1x, where 1 is the slope.) 1) *Substitute 0.5 for the 1 and compare the 2 graphs. (So y= 0.5x or y = x is a child of the parent y = x.) Compare 2) Repeat substituting 0.8, 0.3, 0.1 for the 1in the parent function (y = 1x). Record observations on the table provided. 3) Repeat substituting 2, 3, 5 for the 1 in the parent function (y = 1x). Record observations on the table provided. 4) Repeat substituting -0.3, -0.5, -0.8, -2, -3 for the 1 in the parent function (y = 1x). Record observations on the table provided. 5) Keep the parent graph and delete any other graphs that are stored. This time add the new number 2 to the x (Parent function is y = x or y =x+0; child becomes y = x+2. Compare the parent and the child graphs. Repeat, substituting 3 his time. 6) Keep the parent graph and add the new number  after the x term (Parent function is y = x or y = x+0.) Compare the parent and the child graphs. Repeat, substituting 5 this time and compare again. 7) Keep the parent graph and add the new number - after the x term (Parent function is y = x or y = x+0.) Compare the parent and the child graphs. Repeat, substituting 5 this time and compare again. 8) Keep the parent graph and add the new number  after the x term (Parent function is y = x or y = x+0.) Compare the parent and the child graphs. Repeat, substituting 4 this time and compare again. Project AMP Dr. Antonio Quesada – Director, Project AMP

9) Keep the parent graph and add the new number - after the x term Parent function is y = x or y = x+0 this time and compare again. 10) Record your conclusions on the table.*

D) Plot the parent graph y = x on the graphing calculator. (Remember this is y = 1x. Repeat between the * in Part C for this function. For Parts 5, 6, and 7 consider the parent as y =x+0 and for Parts 8, and 9 consider the parent as y = x +0.

E) Plot the parent graph y = x2 on the graphing calculator. (Remember this is y = (1x)2.) Repeat between the * in Part C for this function. For Parts 5, 6 and 7 consider the parent as y = (x+0)2 and for Parts 8 and 9 consider the parent as y = x2+0.

F) Plot the parent graph y = x3 on the graphing calculator. (Remember this is y = (1 x)3 . Repeat between the * in Part C for this function. For Parts 5, 6, and 7 consider the parent as y =(x+0)3 and for Parts 8 and 9 consider the parent as y = x3+0.

G) Review your work and list any similarities and differences that you observe s you look at all of your work. Project AMP Dr. Antonio Quesada – Director, Project AMP

WORKSHEET Name ______EXPLORATIONS ON y = a(x+h)n + k ______

1. Parent: y = x Children: y = ax, y = x + h, y = x + k

VVariable Parameter Slope (describe line slant) x-intercept y-intercept a >1

0 < a < 1

-1 < a < 0

a < -1

Similarities of Parent & Children Function Differences of Parent & Children Function Project AMP Dr. Antonio Quesada – Director, Project AMP

2. Parent: y = IxI Children: y = a IxI, y = Ix + hI, y = IxI + k

V Variable Parameter Slope (describe line slant) x-intercept y-intercept a >1

0 < a < 1

-1 < a < 0

a < -1

3. Parent: y = x2 Children: y = ax2, y = (x + h)2, y = x2 + k

VVariable Parameter Vertex Opening direction Opening width a >1

0 < a < 1

-1 < a < 0

a < -1

4. Parent: y = x3 Children: y = ax3, y = (x + h)3, y = x3 + k

Variable Parameter Inflection point Width x - intercept a >1

0 < a < 1

-1 < a < 0

a < -1

5. Develop 2 examples using the general function y = a(x + h)n + k for different values of a, h, k, and n ( n = 1, 2, 3). Graph the function on the graphing calculator and state how the parent function was transformed and/or stretched. Find any zeros shown on your graph. (TRACE can help.) Prepare to present each example to the class, including graphs and tables.