Lesson 9

The purpose of this warm-up is to introduce students to the effect of scale factors on the input to a function. They start by sketching the transformation of a graph from a description. Students then write an equation for the new function in terms of the original.

Similar to how \(y=f(x+a)\) is a graph translated left \(a\) units and not right, students may find the relationship between compressing a graph horizontally and the corresponding equation of \(y=f(2x)\) counter-intuitive. This warm-up is intended to address this common misconception directly using the familiar context of distance, time, and speed to help students make sense in their own words why multiplying the input by a scale factor greater than 1 means the graph is compressed instead of stretched.

If students are not sure where to start with the graphs, recommend they make a table for \(t\), \(f(t)\), and \(g(t)\).

The goal of this discussion is for students to compare how they think about the effect of scaling the input to a function with others. Begin the discussion by asking students to share with a neighbor their equation for \(g\) in terms of \(f\) and why their equation makes sense. After brief discussion time, invite students to share whose partner changed their minds about what the equation for \(g\) should be and what their partner said to change their mind. For example, the partner may have reasoned about the graphs using a table of values to see that \(t=2\) for \(f\) is the same as \(t=1\) for \(g\), and so on. They may have also reasoned using the situation, such as how the distance walked after 1 hour in the first graph is now the same as the distance walked after 0.5 hours.

If not brought up by students, tell them that we can say the graph was compressed horizontally by a scale factor of 2 to describe this type of transformation from the graph of \(f\) to the graph of \(g\). Had the scale factor on the input been between 0 and 1, we would have said the graph was stretched horizontally.

If time allows, here are some questions to further the discussion:

• “What would the graph look like if Elena walks 1 mile an hour for the 4 miles?” (The new graph has the same height as \(f\) and the same general shape, but stretched horizontally twice as wide as \(f\) and 4 times wider than \(g\).)
• “Suppose Elena walks her route a different way and that the equation is described by \(h(t)=g(4t)\). What would the graph look like? What could be happening?” (The new graph has the same height as \(f\) and the same general shape, but compressed horizontally so it starts at \((0,0)\) and ends at \((0.5,0)\). The scale factor of 4 would mean Elena travels 4 miles in half an hour, so she's traveling at 8 miles per hour and might be on a bike.)

The goal of this activity is for students to compare the effect of scaling the inputs and outputs of a function. Students consider the changing height above the ground over time for two Ferris wheels whose motions can be described as transformations of the same Ferris wheel, \(F\). This activity purposefully focuses on input-output pairs in tables to encourage students to use specific points as they articulate the differences between the data for the original Ferris wheel and the two new Ferris wheels (MP6).

Monitor for students who have clear explanations for why they cannot determine the value of \(h(20)\) and \(h(60)\) due to the limited amount of information given for \(F(t)\) to share during the discussion.

Arrange students in groups of 2. Give quiet work time for students to complete the table followed by sharing work with a partner and reaching agreement on the values. Once partners agree, they should continue with the rest of the activity.

Some students may think they can figure out the value of \(F(10)\) and \(F(30)\) in order to answer the second question by dividing the value of \(F(20)\) or \(F(60)\) in half. Encourage them to take a close look at the how the output values change with respect to the input values, perhaps by sketching a graph, to help these students understand why they should not assume the data is changing linearly.

The purpose of this discussion is for students to articulate how \(g\) and \(h\) are transformations of \(F\). Begin the discussion by selecting previously identified students to share why there is not enough known about \(F(t)\) to determine the value of \(h(20)\) and \(h(60)\). If any students attempted to estimate the values by plotting the data for \(F\) and sketching out the curve, invite them to share their thinking, but make sure students understand that without knowing the exact value of \(F(10)\) and \(F(30)\), we cannot determine the exact values of \(h(20)\) and \(h(60)\).

Here are some questions for discussion:

• “How is \(g\) related to \(F\)?” (For the same inputs, the outputs of \(g\) are twice as big as the outputs of \(F\) since \(g\) is twice the height of \(F\) and rotates at the same speed.)
• “How did you write an equation for \(g\) in terms of \(F\)?” (Since the output column for \(g(t)\) is twice as large as the output column for \(F(t)\) for the same inputs, I know that \(g(t) = 2F(t)\).)
• “How is \(h\) related to \(F\)?” (For the same outputs, the inputs have to be twice as large since \(h\) is the same size as \(F\) but rotates at half the speed. So instead of taking 20 seconds to go from 212 feet to 0 feet, it takes 40 seconds.)
• “What is the value of \(h(160)\)?” (160 seconds on \(h\) is the same as 80 seconds on \(F\), so \(h(160)=0\).)
• “If there was another Ferris wheel \(j\) where \(j(t)=2F(0.25t)\), what can you say about this wheel?” (\(j\) has twice the diameter of \(F\) and moves a quarter of the speed of \(F\).)

In this activity, students will both write equations to represent transformations shown by graphs and graph transformations described by equations. A piecewise function is used to help students focus on how the inputs or outputs (or both) are affected by scale factors.

Encourage students to use precise language when describing the transformations and to make sure their words match the transformations they describe algebraically (MP6). In particular, students need to be clear whether a scale factor affects the output or the input and if it affects the graph by moving points toward an axis (compressing) or away from an axis (stretching).

Arrange students in groups of 2. Provide access to colored pencils so students can graph \(m\) on the same axes. After 2–3 minutes of quiet work time, tell students to compare their equations for \(k\) and graphs of \(m\). If partners disagree, encourage them to discuss their thinking and work to reach an agreement before moving onto the remaining problems.

Display the graphs of \(P\), \(k\), and \(m\) for all to see. Here are some questions for discussion:

• “How did you figure out the highest point on the graph of \(n\)? What was it?” (Since \(n(x)=P(5x)\), \(n\) has all the same outputs as \(P\), just at different inputs. This means the highest point on \(n\) is \(y=2\), and that happens when \(x=\frac15\).)
• “Name another transformation we could do to \(P\) where the resulting graph would have a highest point at 2.” (Any horizontal scale factor or horizontal translation. \(y=\text-P(x)\) also has a highest point at 2 since \(y=P(x)\) has a lowest point at -2.)

If time allows, conclude the discussion by challenging students to come up with at least two possible equations for \(q\). (\(q(x)=P(\frac{4}{18}x)\), \(q(x)=P(\text-\frac{1}{6}x)\), \(q(x)=P(x-12)\), and \(q(x)=P(\frac14x-2)\) are 4 possibilities.) Invite students to share their ideas, recording responses for all to see.