Terry Guay Lesson Plan: Mathematics
Course Title: Algebra 2
Grade Level: 10
Unit Title: Graphs of trigonometric functions
Ohio Benchmark: Show and describe the results of combinations of translations, reflections
and rotations (compositions); e.g., perform compositions and specify the result of a composition
as the outcome of a single motion, when applicable.
Geometry and Spatial Sense Standard, Benchmark 9. Grade 10. Page 174.
Academic Content Standards for Mathematics K-12, Ohio Department of Education
http://www.ode.state.oh.us/GD/Templates/Pages/ODE/ODEDetail.aspx?page=3&TopicRelationI
D=1704&ContentID=801&Content=50719
Goal: Students will learn how to translate the graph of a sine function using the phase shift and
the vertical shift.
Behavioral Objective: Students will correctly graph 4 translated sine functions with 75%
accuracy.
Gaining Attention
I will introduce the concept of the sine function and its graph.
 What do you know about the sine function?
 What is the sine function used for?
(4. Questioning, 5. Learner experience)
(open ended questions. Expect answers to include sine = opposite/hypotenuse,
used in right triangles, can be used to find missing angle or missing side, is the y
value in a coordinate pair on the unit circle)
Informing the learner of the objective
Learners will be expected to correctly graph a translated sine function.
Use advance organizer to show different variations on the graph of the function y = sin x.
(1. Content)
Stimulating recall of prerequisite learning
Begin review the graph of the parent sine function y = sin x:
1. Allow students to work in groups of 2 or 3 to sketch and analyze the graph of y =
sin x. Have students use graph paper. They may use degrees or radians. Remind
students that they know the coordinates of points for the graph from the unit
circle. Ask students to make a list of characteristics of the graph.
(5. Learner experience, 7. Discussion, Cooperative Learning Small Group Task)
2. Whole class discussion: Gather information from the groups to make a list on the
board of characteristics of the graph of the function y = sin x.
Who can name a characteristic of the graph of the sine function?
(4. Questioning, 5. Learner experience, 7. Discussion)
(expect answers to include: starts at 0, goes up first, then back to 0, then goes
down, then back up to 0, then repeats. Goes on forever to left and right. Repeats
every 2 pi or every 360 degrees, graph height goes up and down 1 unit from the x
axis, has a maximum of 1, has a minimum of -1)
Presenting the stimulus material
I will explain that the sine function can be modified and the graphs of the modified
functions can be predicted from the form of the equation. Also, equations of functions
can be determined from their graphs.
Group activity: Groups of 3 students will be asked to complete the activity “Translating
Graphs of Sine Functions – Group Activity – Inductive.” In each group, one student will
be the coordinator (keep the group on task, asks questions, directs the group members to
do their tasks), one student will be the recorder (writing answers on the worksheet), one
student will be the tech person (works the calculator).
(7. Discussion, Inquiry Learning, Cooperative Learning Small Group Task)
Inductive Learning or Deductive Learning
Inductive Learning – Students are asked to use their calculators to graph
functions given to them which have variations in the phase shift (horizontal
translation) and vertical translation and make inferences about how the values in
the function equation affect the graph. See “Translating Graphs of Sine Functions
– Group Activity – Inductive.”
Deductive Learning – Students are asked to hypothesize how the phase
shift and vertical translation numbers (h, b) affect the graph of the function. Then
they are to confirm or refute their hypotheses using their calculators and functions
given to them which have variations in horizontal and vertical translations. See
“Translating Graphs of Sine Functions – Group Activity – Deductive.”
(2. Conceptual Movement, 3. Examples and Nonexamples, 5. Learner
Experience)
Differentiated Instruction:
Lower Level: Direct students to look at individual points on the original
sine function and the function with a translation and compare what the y value is
or the x value is. Guide them to see the relationship between the phase shift or
horizontal shift number in the function given with the movement of the graph.
Advanced: Ask students to write an equation for a sine function with
given horizontal and/or vertical translations.
Eliciting the desired behavior
Group discussion (whole class): Ask groups to report their findings from the Group
Activity.
 Who can explain the movement of the graph y = sin x when it becomes
y = sin x + b?
How do we know what direction the graph shifts?
Does the period change? Does the amplitude change?
What are the new max and min values with a shift of +1 unit.
(expect answers to include: a positive number b shifts the graph up by b units. A
negative number b shifts the graph down by b units. No period change, no
amplitude change. New max=2 and min = 0 with a shift of +1 unit)
(4. Questioning, 5. Learner Experience, 7. Discussion, Concept Learning)

Who can explain the movement of the graph y = sin x when it becomes
y = sin (x – h)?
How do we know what direction the graph shifts?
Does the period change?
Where does the graph begin and end one cycle for a shift y = sin (x-pi/2)?
Does the amplitude change? Do the max and min change?
(expect answers to include: a positive number b shifts the graph up by b units. A
negative number b shifts the graph down by b units. No period change, no
amplitude change. New max=2 and min = 0 with a shift of +1 unit)
(4. Questioning, 6. Student Self-evaluation, 7. Discussion, Concept Learning)
Providing Feedback
Feedback is given as students in the class discuss their analysis of translating functions.
Feedback will be in the form of confirmation, quick and firm, or positive verbal prompts
when cues or hints or corrections are needed. When a student is hesitant but correct, I
will confirm that they are correct and restate what they have just stated correctly. When
students show lack of knowledge, I will give cues and use guiding questions to help the
student toward discovery of the concepts. Together we can discover the obstacle they are
having so they can be successful. Sometimes a simple question will guide them to
figuring out for themselves what they have already seen or know what to try next to be
able to come to a conclusion. If a student is having great difficulty, I will work with the
student one-on-one with examples and nonexamples.
(4. Questioning, 2. Conceptual Movement, 3. Examples and nonexamples, 6. Student
Self-evaluation, 7. Discussion)
Assessing the behavior
I will provide a quiz with four problems asking students to sketch a translated sine
function graph.
Advanced: Ask students to write an equation for a sine function with given horizontal
and/or vertical translations.
Algebra 2 Advance Organizer
The Sine Function Graph
Name________________________
Period____Date________________
Amplitude
a
Sine Function
Period
2π/k
y = sin (x)
y = a sin k (x–h) + b
Vertical
Shift
b
Phase Shift
(Horizontal
Translation)
h
Algebra 2
The Sine Function Graph
Group Activity
1)
Name________________________
Name________________________
Period____Date________________
Graph the function y = sin (x)
.
2) What are some characteristics you notice about the graph (size, shape, location, max, min,
start, end, etc.)?
____________________________________________________________________________
____________________________________________________________________________
____________________________________________________________________________
____________________________________________________________________________
____________________________________________________________________________
____________________________________________________________________________
____________________________________________________________________________
____________________________________________________________________________
____________________________________________________________________________
____________________________________________________________________________
____________________________________________________________________________
Algebra 2
Translating Graphs of Sine Functions
Group Activity - Inductive
Coordinator ___________________
Recorder _____________________
Tech ________________________
Period____Date________________
Using your calculator in radian mode with the trig window (zoom 7), graph the function.
1) y = sin (x)
.
2) y = sin (x) + 2
.
3) y = sin (x) – 3
4) y = sin (x) + 1
.
Explain the movement of the graph y = sin x when it becomes y = sin x + b. _______________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Using your calculator in radian mode with the trig window (zoom 7), graph the function.
5) y = sin (x – π)
.
6) y = sin (x + π/2)
.
7) y = sin (x + 2π/3)
8) y = sin (x – π/4)
.
Explain the movement of the graph y = sin x when it becomes y = sin (x – h). _______________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Algebra 2
Translating Graphs of Sine Functions
Group Activity - Deductive
Coordinator ___________________
Recorder _____________________
Tech ________________________
Period____Date________________
1) Using your calculator in radian mode with the trig window (zoom 7), graph the function.
y = sin (x)
.
B. Make a conjecture about how this graph will look: y = sin (x) + 2
A. My hypothesis is: _______________________________________________________
______________________________________________________________________
______________________________________________________________________
B. Graph the function on the calculator and record below.
.
C. Was your hypothesis correct? _______ If not, how could you correct your hypothesis?
____________________________________________________________________________
____________________________________________________________________________
3) Make a conjecture about how this graph will look: y = sin (x) – 3
A. My hypothesis is: _______________________________________________________
______________________________________________________________________
______________________________________________________________________
B. Graph the function on the calculator and record below.
C. Was your hypothesis correct? _______ If not, how could you correct your hypothesis?
____________________________________________________________________________
____________________________________________________________________________
4) Make a conjecture about how this graph will look: y = sin (x) + 1
A. My hypothesis is: _______________________________________________________
______________________________________________________________________
______________________________________________________________________
B. Graph the function on the calculator and record below.
C. Was your hypothesis correct? _______ If not, how could you correct your hypothesis?
____________________________________________________________________________
____________________________________________________________________________
Summary: Explain the movement of the graph y = sin x when it becomes y = sin x + b.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Using your calculator in radian mode with the trig window (zoom 7), graph the function.
5) Make a conjecture about how this graph will look: y = sin (x – π)
A. My hypothesis is: _______________________________________________________
______________________________________________________________________
______________________________________________________________________
B. Graph the function on the calculator and record below.
.
C. Was your hypothesis correct? _______ If not, how could you correct your hypothesis?
____________________________________________________________________________
____________________________________________________________________________
6) Make a conjecture about how this graph will look: y = sin (x + π/2)
A. My hypothesis is: _______________________________________________________
______________________________________________________________________
______________________________________________________________________
B. Graph the function on the calculator and record below.
.
C. Was your hypothesis correct? _______ If not, how could you correct your hypothesis?
____________________________________________________________________________
____________________________________________________________________________
7) Make a conjecture about how this graph will look: y = sin (x + 2π/3)
A. My hypothesis is: _______________________________________________________
______________________________________________________________________
______________________________________________________________________
B. Graph the function on the calculator and record below.
.
C. Was your hypothesis correct? _______ If not, how could you correct your hypothesis?
____________________________________________________________________________
____________________________________________________________________________
8) Make a conjecture about how this graph will look: y = sin (x – π/4)
A. My hypothesis is: _______________________________________________________
______________________________________________________________________
______________________________________________________________________
B. Graph the function on the calculator and record below.
.
C. Was your hypothesis correct? _______ If not, how could you correct your hypothesis?
____________________________________________________________________________
____________________________________________________________________________
Summary: Explain the movement of the graph y = sin x when it becomes y = sin (x – h).
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________