Lesson 5
Some Functions Have Symmetry
5.1: Changing Heights (5 minutes)
Warm-up
This warm-up invites students to use their understanding of a the motion of a Ferris wheel to complete a table of values where the data has distinct symmetry.
Student Facing
The table shows Clare’s elevation on a Ferris wheel at different times, \(t\). Clare got on the ride 80 seconds ago. Right now, at time 0 seconds, she is at the top of the ride. Assuming the Ferris wheel moves at a constant speed for the next 80 seconds, complete the table.
| time (seconds) | height (feet) |
|---|---|
| -80 | 0 |
| -60 | 31 |
| -40 | 106 |
| -20 | 181 |
| 0 | 212 |
| 20 | |
| 40 | |
| 60 | |
| 80 |
Student Response
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Activity Synthesis
Select students to explain how they completed the table. If not brought up by students, ask, "What does the graph of this function look like?" (A curve that is symmetric about the vertical axis.) Invite students to share their descriptions or any graphs they make of the data. If no students sketched a graph that can be shared, display one for all to see to help highlight the symmetry.
5.2: Card Sort: Two Types of Graphs (15 minutes)
Activity
The purpose of this activity is for students to identify common features in graphs of functions that are the same when reflected across the vertical axis and those that look the same when reflected across both axes. This leads to defining these types of functions as even or odd, respectively. In the following activity, students match tables to the graphs and refine their definitions, so groups should keep their copies of the blackline master from this activity to use in the next activity.
Monitor for different ways groups choose different categories, but especially for categories that distinguish between graphs of even functions and graphs of odd functions. As students work, encourage them to refine their descriptions of the graphs using more precise language and mathematical terms (MP6).
Launch
Arrange students in groups of 2. Tell them that in this activity, they will sort some cards into categories of their choosing. When they sort the graphs, they should work with their partner to come up with categories. Distribute pre-cut slips to each group.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Supports accessibility for: Organization; Attention; Social-emotional skills
Student Facing
Your teacher will give you a set of cards that show graphs. Sort the cards into 2 categories of your choosing. Be prepared to explain the meaning of your categories.
Student Response
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Anticipated Misconceptions
Some students may focus too closely on identifying specific points on the graph to use to make their categories. Encourage these students to look at the graph as a whole while they sort.
Activity Synthesis
Select groups to share their categories and how they sorted their graphs. Choose as many different types of categories as time allows, but ensure that one set of categories distinguishes between graphs of even functions and graphs of odd functions. Attend to the language that students use to describe their categories, giving them opportunities to describe the types of graphs more precisely.
It is possible students will think of graphs of odd functions as ones where a \(180^{\circ}\) rotation using the origin as the center of rotation results in the same graph. While it is true that this type of rotation appears the same as successive reflections of the graph across both axes, focus the conversation on thinking in terms of reflections since the function notation students will use to describe odd functions, \(g(x)=\text-g(\text-x)\), algebraically describes reflections.
At the conclusion of the sharing, display the graphs of the even functions next to the odd functions. Tell students that functions whose graphs look the same when reflected across the \(y\)-axis are called even functions. Functions whose graphs that look the same when reflected across both axes are called odd functions. In the next activity, students refine their understanding of even and odd functions by pairing each of the graphs with a table of values and writing their own description for these two types of functions based on inputs and outputs.
5.3: Card Sort: Two Types of Coordinates (15 minutes)
Activity
The purpose of this activity is for students to deepen their understanding of even and odd functions. Using the graphs from the previous activity, students first match each graph to a table of coordinate pairs and then use both representations to identify defining characteristics of even functions and odd functions (MP8). In the next lesson, students will learn how to use an equation to prove if a function is even or odd, so an important result of this activity is describing even and odd functions using function notation.
Monitor for students making connections between the transformations described in the previous activity (a reflection across the \(y\)-axis versus successive reflections across both axes) and the coordinates in the tables to share during the whole-class discussion.
Launch
Keep students in the same groups. If students do not already have their slips from the previous activity arranged into two groups, one for graphs of even functions and one for graphs of odd functions, ask them to do so now. Distribute pre-cut slips.
Design Principle(s): Support sense-making; Maximize meta-awareness
Supports accessibility for: Organization; Attention; Social-emotional skills
Student Facing
Your teacher will give you a set of cards to go with the cards you already have.
- Match each table of coordinate pairs with one of the graphs from earlier.
- Describe something you notice about the coordinate pairs of even functions.
- Describe something you notice about the coordinate pairs of odd functions.
Student Response
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Student Facing
Are you ready for more?
- Can a non-zero function whose domain is all real numbers be both even and odd? Give an example if it is possible or explain why it is not possible.
- Can a non-zero function whose domain is all real numbers have a graph that is symmetrical around the \(x\)-axis? Give an example if it is possible or explain why it is not possible.
Student Response
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Activity Synthesis
The goal of this discussion is for students to move from observations about specific even and odd functions to generalizing things that are true for all even functions and for all odd functions.
Select previously identified students to share the connections they see between the transformations associated with graphs of even and odd functions and features of the coordinate pairs belonging to even and odd functions. Record these observations for all to see in two lists: one for even, one for odd.
If time allows, assign groups to write a single sentence describing even or odd functions that summarizes one of the lists.
Supports accessibility for: Visual-spatial processing
Lesson Synthesis
Lesson Synthesis
The goal of this discussion is for students to use function notation to summarize their understanding of what makes a function even and what makes a function odd.
Here are some questions for discussion to help students transition to using function notation:
- “Using the the language of inputs and outputs, what is true about even functions?” (Opposite inputs have the same output.)
- “Using the the language of inputs and outputs, what is true about odd functions?” (Opposite inputs have opposite outputs.)
- “If a function \(f\) is even and \(f(3)=7\), what is something else you know about \(f\)?” (Since \(f\) is even, if an input of 3 has an output of 7, then an input of -3 also has an output of 7, so \(f(\text-3)=7\).)
- “If a function \(g\) is odd and \(g(5)=\text-1\), what is something else you know about \(g\)?” (Since \(g\) is odd, if an input of 5 has an output of -1, then an input of -5 has an output of 1, so \(g(\text-5)=1\).)
Tell students that these observations can be generalized for all even and odd functions. If a function \(f\) is even, then \(f(x)=f(\text-x)\) is true. If a function \(g\) is odd, then \(g(x)=\text-g(\text-x)\). Students will focus on these definitions in the next lesson.
5.4: Cool-down - Even or Odd? (5 minutes)
Cool-Down
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Student Lesson Summary
Student Facing
We've learned how to transform functions in several ways. We can translate graphs of functions up and down, changing the output values while keeping the input values. We can translate graphs left and right, changing the input values while keeping the output values. We can reflect functions across an axis, swapping either input or output values for their opposites depending on which axis is reflected across.
For some functions, we can perform specific transformations and it looks like we didn't do anything at all. Consider the function \(f\) whose graph is shown here:
What transformation could we do to the graph of \(f\) that would result in the same graph? Examining the shape of the graph, we can see a symmetry between points to the left of the \(y\)-axis and the points to the right of the \(y\)-axis. Looking at the points on the graph where \(x=1\) and \(x=\text-1\), these opposite inputs have the same outputs since \(f(1)=4\) and \(f(\text-1)=4\). This means that if we reflect the graph across the \(y\)-axis, it will look no different. This type of symmetry means \(f\) is an even function.
Now consider the function \(g\) whose graph is shown here:
What transformation could we do to the graph of \(g\) that would result in the same graph? Examining the shape of the graph, we can see that there is a symmetry between points on opposite sides of the axes. Looking at the points on the graph where \(x=1\) and \(x=\text-1\), these opposite inputs have opposite outputs since \(g(1)=2.35\) and \(g(\text-1)=\text-2.35\). So a transformation that takes the graph of \(g\) to itself has to reflect across the \(x\)-axis and the \(y\)-axis. This type of symmetry is what makes \(g\) an odd function.