Unit 5, Lesson 4
Reflecting Functions
Integrated Math 3
4.1
Warm-up
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
The purpose of this Warm-up is to elicit the idea that graphs can be reflected both vertically and horizontally, which will be useful when students generate a table and graph for and from in a later activity. Students should be familiar with the properties of reflections from a previous course. While students may notice and wonder many things about these images, being precise about how the graphs are related is the important discussion point.
This prompt gives students opportunities to see and make use of structure (MP7). They should notice that the distance of each point on the graph to the line of reflection is the same but in a different direction after the graph is reflected. In this unit, reflections across the - and -axes are the focus.
Launch
Display the three graphs for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.
Activity
NoneWhat do you notice? What do you wonder?
Graph of function g on a coordinate plane. X axis from negative 3 to 5. Y axis from negative 4 to 4. From left to right, the function begins around negative 3 comma 2, moves downward and to the right to about negative 1 comma negative 2, stays level until about 2 comma negative 2, moves upward and to the right to about 3 comma 2, moves downward and to the right to 4 comma negative 1, then stays level until 5 comma negative 1.
Graph of function h on a coordinate plane. X axis from negative 5 to 3. Y axis from negative 4 to 4. From left to right, the function begins around negative 5 comma 1, moves to the right to about negative 4 comma 1, moves downward and to the right to about negative 3 comma negative 2, moves upward and to the right to about negative 2 comma 2, stays level until about 1 comma 2, then moves downward and to the right until about 3 comma negative 2.
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses. If possible, record the relevant reasoning on or near the graphs. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If lines of reflection do not come up during the conversation, ask students to discuss this idea. Students will focus on what values do and do not change when reflecting across the - and -axes during this lesson, so it is okay if students do not focus on specific coordinate values at this time.
Lesson Synthesis
The purpose of this discussion is for students to describe the effects of reflections and translations of a function to its graph and equation.
Display the image and prompt:
Describe how to transform the graph of to get the graph of:
After some quiet think time, invite students to share their responses. Sample responses:
- Reflect across the -axis and translate up 4 units.
- Translate to the right 7 units and then reflect across the -axis.
- Reflect across the -axis and then reflect across the -axis.
If time allows, highlight for students that for these 3 graphs, the order in which they complete the transformations doesn't matter. For example, if we reflect across the -axis and then translate up 4 units or we translate the graph of up 4 units and then reflect it across the -axis, we get the same graph. Invite students to think of a transformation with at least two moves where the order does matter. (Translating up 2 and reflecting across the -axis is different from reflecting across the -axis and then translating up 2.)
Student Lesson Summary
Here are graphs of the functions , , and , where and . How do these equations match the transformation we see from to and from to ?
A graph of function f on a coordinate plane. X axis from negative 3 to 3, by 1’s. Y axis from negative 3 to 3, by 1’s. From left to right, the function begins in the third quadrant, moves upward and to the right, crossing the x axis at approximatelynegative 1, continues upward reaching about 0 comma 1. It then curves back downward slightly then back up and to the right, and continues to move upward and to the right, ending in the first quadrant.
A graph of function g on a coordinate plane. X axis from negative 3 to 3, by 1’s. Y axis from negative 3 to 3, by 1’s. From left to right, the function begins in the second quadrant, moves down and to the right crossing the x axis at about negative 1. It moves downward and to the right crossing the y axis at about negative 1 then moves downward and to the right passing through about 1 point 5 comma negative 2 point 5, then ending in the fourth quadrant.
A graph of function h on a coordinate plane. X axis from negative 3 to 3, by 1’s. Y axis from negative 3 to 3, by 1’s. From left to right, the function begins in the second quadrant, moves down and to the right to about negative 1 comma 1. It crosses the y axis at about 1 then moves downward and to the right passing through the x axis at about 1, then ending in the fourth quadrant.
Considering first the equation , we know that for the same input , the value of will be the opposite of the value of . For example, since , we know that . We can see this relationship in the graphs where is the reflection of across the -axis.
Looking at , this equation tells us that the two functions have the same output for opposite inputs. For example, 1 and -1 are opposites, so (and is also true!). We can see this relationship in the graphs where is the reflection of across the -axis.