Unit 6, Lesson 4
Reflecting Functions
Algebra 2
4.1
Warm-up
Notice and Wonder: Reflections
Instructional Routines
Notice and Wonder
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Materials
None
Activity 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
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
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
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
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
Student Lesson Summary
Here are graphs of the functions
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
Looking at
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