Identify the effect on the graph of replacing by , , , and for specific values of (both positive and negative); find the value of given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
This Warm-up prompts students to compare four functions and their graphs. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the functions for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three functions that go together and can explain why. Next, tell students to share their response with their group and then work together to find as many sets of three as they can.
Activity
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Which three go together? Why do they go together?
A
B
C
D
Student Response
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Building on Student Thinking
Activity Synthesis
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, such as "reflection," "translation," "input," and "output," and to clarify their reasoning as needed. Consider asking:
In this activity, students compare how the same transformation affects different functions. As students create equations for different functions that have undergone the same transformations using their graphs, they have an opportunity to leverage repeated reasoning to notice patterns in the equations (MP8).
This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.
Launch
Arrange students in groups of 4. Assign each group one of these original functions:
In this activity, students graph a transformation, then trade with a partner. As students identify the original function and transformation from a graph, they are reasoning about the affects of transformations on a graph. This activity also gives students additional practice with graphing transformations of functions and makes use of repeated reasoning as students begin to generalize about transformations of functions (MP8).
Launch
Arrange students in groups of 2. Each student chooses a different function and transformation.
If time allows, invite students to choose an additional set and repeat the process.
Engagement: Develop Effort and Persistence. Check in and provide each group with feedback that encourages collaboration and community. For example, highlight characteristics of the graph for groups to discuss. Supports accessibility for: Social-Emotional Functioning, Organization
Activity
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Student Lesson Summary
When we apply transformations to functions, there are some patterns that we can see regardless of what type of function we started with. For example, let's apply this transformation to two different functions: shift left 3 and stretch vertically by a factor of 2.
Let's start with an original function . Here are graphs of the original function and the transformed function:
We can see that the vertex has shifted left to the point . We can also see that the slope of the piece that is decreasing is now -2, and the slope of the piece that is increasing is 2. This means we can write an equation for the transformed function: .
Now let's try a different original function, , but use the same transformation. Here are graphs of this original function and the transformed function:
We can see that the vertex of this function has also shifted left to the point . The graph is narrower than the original because of the vertical stretch. An equation for this function is: .
The transformation changed each original function in the same ways: shifting left 3 and then stretching. This means we can make some observations about the changes in the equations. When the transformation affects the graph horizontally, such as a horizontal shift, reflection, or stretch, then it affects the input of the function. The changes to the equation will be grouped with the input as well. When the transformation affects the graph vertically, such as vertical shift, reflection, or stretch, then it affects the output of the function.
This makes sense, since the horizontal axis represents the input of the function, and the vertical axis represents the output of the function.
to access Student Response.
Activity
None
Your teacher will assign you an original function. Create a display that includes each of these steps:
Graph the original function in the coordinate plane. Plot and label the vertical intercept.
Transform the function by a translation down 3 and horizontal stretch by a factor of 2.
Graph this transformed function in the same coordinate plane using a different color.
Identify and label the image of the point from the original function onto the transformed function.
Write the equation of the transformed function.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is for students to notice patterns in the graphs and equations for each function. Invite students to inspect each others' displays. After students have compared their displays with the other function types, use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to this question: "How do you know whether the transformations affect the input or the output from the equation?" In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner’s ideas and writing. Display the prompt, then give students 1–2 minutes to write a first draft before pairing with others.
Display these prompts for feedback:
“ makes sense, but what do you mean when you say ?”
“Can you describe that another way?”
“How do you know ? What else do you know is true?”
Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.
If time allows, here are some additional questions to ask:
“Where do you see the horizontal stretch in each equation?” (The coefficient of was for each equation.)
“How did the horizontal stretch change the shape of each graph?” (In each case, the graph of the transformed function was twice as wide.)
“Where do you see the translation in each equation?” (There is a constant term of -3.)
“How did the translation change each graph? What about the -intercept?” (Each graph was shifted down 3, including the intercepts. For the first 3 functions, this moved the -intercept to , but since the exponential function had a different original -intercept, the new -intercept was at .
“How do you know whether the transformations affect the input or the output from the equation?” (If the transformation is vertical, like vertical translation, reflection, or stretch, then it affects the output. If the transformation is horizontal, like horizontal translation, reflection, or stretch, then it affects the input, so it will be in the parentheses with the .)
Each person will choose a function from Column A and a transformation from Column B. Don't tell your partner what you chose!
Graph the transformed function on the grid paper.
When you and your partner have both drawn your functions, trade papers.
Identify the original function and transformation that your partner chose.
Write the equation of the transformed function on the graph, then take turns explaining to your partner how you know.
Column A
Column B
Reflect over the -axis and translate up 4.
Horizontal stretch by a factor of 3 and reflect over the -axis.
Translate left 5 and a vertical compression by a factor of .
Horizontal stretch by a factor of 2 and vertical stretch by a factor of 2.
Student Response
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Building on Student Thinking
If students struggle to identify the original function that their partner used after trading papers, consider saying:
“Tell me more about what features of the graph stand out to you.”
“How could looking at some of the displays made in the previous activity help you identify the original function and transformation?”
Activity Synthesis
The purpose of this discussion is for students to generalize about transformations of graphs of functions, as well as observe that not all transformations are obvious from the graph alone. Here are some questions to elicit discussion:
“What transformations were hard to identify from the graph?”
“When did the order of transformations make a difference?” (When doing a stretch and shift in the same direction. For example, a vertical stretch and then vertical shift has a different outcome than a vertical shift and then vertical stretch.)
Building Toward
Standards Alignment
Building On
Addressing
HSF-BF.B.3
Identify the effect on the graph of replacing by , , , and for specific values of (both positive and negative); find the value of given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Identify the effect on the graph of replacing by , , , and for specific values of (both positive and negative); find the value of given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
