Lesson 4

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 \(\text-f(x)\) and \(f(\text-x)\) from \(f(x)\) 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 \(x\)- and \(y\)-axes are the focus.

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.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. 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 \(x\)- and \(y\)-axes during this lesson, so it is okay if students do not focus on specific coordinate values at this time.

Building on the warm-up, in this activity students compare two functions where one is a reflection of the other across the \(x\)-axis. They first use coordinates on the graph of \(f\) to complete a table and compute \(g(x)=\text-f(x)\) for each \(x\)-value in the table. Students then plot \(g\) on the same axes as \(f\), allowing them to directly compare the graphs of \(f\) and \(g\) before explaining why \(g(x) = \text-f(x)\) means the graph of \(g\) is a reflection of the graph of \(f\) across the \(x\)-axis.

Display the graph of function, \(f\), from the Task Statement and explain that a different function, \(g\), is defined by \(g(x)=\text-f(x)\). Ask students, “What is \(f(\text-1)\)? \(g(\text-1)\)?” After some quiet think time, invite students to share their responses and explain how they determined their answer. Make sure the discussion includes the key idea that the point \((\text-1,\text-4)\) is on the graph of \(f\) and the point \((\text-1,4)\) is on the graph of \(g\).

Students may think that the two graphs are related by a different type of transformation: reflection over the \(y\)-axis and 180 degree rotation around \((0.6,0)\) are both reasonable guesses. Have these students (and any other students questioning if their answer is correct) use tracing paper to check their initial guesses.

Invite students to share their explanation of how the graph of \(g\) is related to the graph of \(f\) and why. The key idea in this discussion is that \(g(x) = \text-f(x)\) produces a reflection across the \(x\)-axis by holding the inputs constant and taking the outputs to their opposites. Students may recall from geometry that in a reflection, the line of symmetry (the \(x\)-axis here) makes a perpendicular bisector when connecting a point to its image, but this level of detail is not necessary.

Continuing the thinking from the previous activity with the same starting function, students now compare two functions where one is a reflection of the other across the \(y\)-axis. Keeping the same activity structure, they first use the coordinates on the graph of \(f\) to complete a table and compute \(h(x) = f(\text-x)\) for corresponding points in the table. Unlike the previous activity, students fill in both \(x\)- and \(y\)-coordinates for the function \(h\). They then plot \(h\) on the same axes as \(f\) to compare the graphs directly before explaining why \(h(x) = f(\text-x)\) means the graph of \(h\) is a reflection of the graph of \(f\) across the \(y\)-axis.

Arrange students in groups of 2. Display the graph of function \(f\) and the prompt:

• Let \(h\) be the function defined by \(h(x) = f(\text-x)\). Make a prediction about what the graph of \(h\) will look like.

Allow students 1 minute of quiet think time, then invite students to briefly discuss their predictions with their partner. Tell students that they will confirm their prediction in the activity.

Students may think that the two graphs are related by a different type of transformation: reflection over the \(x\)-axis and 180 degree rotation around \((0.6,0)\) are both reasonable guesses. Have these students (and any other students questioning if their answer is correct) use tracing paper to check their initial guesses.

Invite students to share their explanation of how the graph of \(h\) is related to the graph of \(f\) and why. The key idea in this discussion is that \(h(x) = f(\text-x)\) produces a reflection across the \(y\)-axis by holding the output values constant and taking the input values to their opposites.