# Lesson 9

Scaling the Inputs

## 9.1: Out and Back (5 minutes)

### Warm-up

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.

### Student Facing

Every weekend, Elena takes a walk along the straight road in front of her house for 2 miles, then turns around and comes back home. Let’s assume Elena walks at a constant speed.

Here is a graph of the function $$f$$ that gives her distance $$f(t)$$, in miles, from home as a function of time $$t$$ if she walks 2 miles per hour.

1. Sketch a graph of the function $$g$$ that gives her distance $$g(t)$$, in miles, from home as a function of time $$t$$ if she walks 4 miles per hour.

2. Write an equation for $$g$$ in terms of $$f$$. Be prepared to explain why your equation makes sense.

### Anticipated Misconceptions

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

### Activity Synthesis

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.)

## 9.2: A New Set of Wheels (15 minutes)

### Activity

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.

### Launch

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.

Representation: Internalize Comprehension. Activate or supply background knowledge. Provide students with access to a copy of the picture of the Ferris wheel and table from an earlier lesson to support information processing by providing a visual.
Supports accessibility for: Visual-spatial processing; Organization

### Student Facing

Remember Clare on the Ferris wheel? In the table, we have the function $$F$$ which gives her height $$F(t)$$ above the ground, in feet, $$t$$ seconds after starting her descent from the top. Today Clare tried out two new Ferris wheels.

• The first wheel is twice the height of $$F$$ and rotates at the same speed. The function $$g$$ gives Clare's height $$g(t)$$, in feet, $$t$$ seconds after starting her descent from the top.
• The second wheel is the same height as $$F$$ but rotates at half the speed. The function $$h$$ gives Clare's height $$h(t)$$, in feet, $$t$$ seconds after starting her descent from the top.
$$t$$ $$F(t)$$ $$g(t)$$ $$h(t)$$
0 212
20 181
40 106
60 31
80 0
1. Complete the table for the function $$g$$.
2. Explain why there is not enough information to find the exact values for $$h(20)$$ and $$h(60)$$.
3. Complete as much of the table as you can for the function $$h$$, modeling Claire's height on the second Ferris wheel.
4. Express $$g$$ and $$h$$ in terms of $$f$$. Be prepared to explain your reasoning.

### Anticipated Misconceptions

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.

### Activity Synthesis

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 $$F$$.)
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their explanations as to why there is not enough information to calculate $$h(20)$$ and $$h(60)$$, present an incorrect answer and explanation. For example, “I can find $$h(20)$$ by taking half the difference of $$F(0)$$ and $$F(20)$$ and subtracting it from 212. So $$h(20)$$ is 196.5.” Ask students to discuss with a partner what the error is in the explanation, critique the author’s reasoning, and write a correct explanation. As students discuss with a partner, listen for students who refer to the changes in the output values with respect to the input values. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to describe the transformations between $$F(t)$$, $$g(t)$$, and $$h(t)$$. This helps students evaluate and improve on the written mathematical arguments of others as they compare the effects of scaling the inputs and outputs of a function.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

## 9.3: The Many Transformations of a Function $P$ (15 minutes)

### Activity

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).

### Launch

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.

Speaking: MLR8 Discussion Supports. Provide sentence frames to support students in producing statements describing the similarities and differences between their equations for $$k$$ and graphs of $$m$$. For example, “I disagree because….”, “We can agree that….” or “What makes _____ different from _____ is _____ .”
Design Principle(s): Support sense-making; Optimize output (for comparison)
Action and Expression: Internalize Executive Functions. Provide students with a two-column table to record points and identify patterns on function $$k$$ and its relationship to function $$P$$.
Supports accessibility for: Language; Organization

### Student Facing

Function $$k$$ is a transformation of function $$P$$ due to a scale factor.

1. Write an equation for $$k$$ in terms of $$P$$.
2. On the same axes, graph the function $$m$$ where $$m(x)=P(0.75x)$$.
3. The highest point on the graph of $$P$$ is $$(1,2)$$. What is the highest point on the graph of a function $$n$$ where $$n(x)=P(5x)$$? Explain or show your reasoning.
4. The point furthest to the right on the graph of $$P$$ is $$(4,0)$$. If the point furthest to the right on the graph of a function $$q$$ is $$(18,0)$$, write a possible equation for $$q$$ in terms of $$P$$.

### Student Facing

#### Are you ready for more?

What transformation takes $$f(x)=2x(x-4)$$ to $$g(x)=8x(x-2)$$?

### Activity Synthesis

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.

## Lesson Synthesis

### Lesson Synthesis

Arrange students in groups of 2. Display the following for all to see:

$$f(x)=(x+3)(x-2)$$

1. $$h(x)=4(x+3)(x-2)$$
2. $$j(x)=(0.5x+3)(0.5x-2)$$
3. $$k(x)=(3x+3)(3x-2)-5$$

Tell students that the goal is for them to identify what transformation would take the graph of $$f$$ to the graph of the new function ($$h$$: stretch vertically by a factor of 4, $$j$$: stretch horizontally by a factor of 2, $$k$$: compress horizontally by a factor of $$\frac13$$ and translate down 5 units). Give students brief quiet think time, and then ask them to share with a partner. Tell students that if they disagree with their partner's description, they should discuss their thinking and work to reach an agreement. Select students to share their descriptions, recording for all to see. If not brought up by students, emphasize that scaling the input of a function means horizontal movement away from or toward the vertical axis while scaling the output of a function means vertical movement away from or toward the horizontal axis.

## 9.4: Cool-down - The Right Scale (5 minutes)

### Cool-Down

Here are two graphs showing the distance traveled by two trains $$t$$ hours into their journeys. What do you notice?
A train travelling twice the speed gets to any particular point along the track in half the time, so the graph for Train B is compressed horizontally by a factor of $$\frac12$$ when compared to the graph of Train A. If the function $$f(t)$$ represents the distance Train A travels in $$t$$ hours, then $$f(2t)$$ represents the distance Train B travels in $$t$$ hours, because Train B goes as far in $$t$$ hours as Train A goes in $$2t$$ hours.
If a different Train C were going one fourth the speed of Train A, then its motion would be represented by $$s = f(0.25t)$$ and the graph would be stretched horizontally by a factor of 4 since it would take four times as long to travel the same distance.