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 is a graph translated left units and not right, students may find the relationship between compressing a graph horizontally and the corresponding equation of 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.

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 in terms of and why their equation makes sense. After brief discussion time, invite students to share whose partner changed their minds about what the equation for 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 for is the same as for , 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 to the graph of . Had the scale factor on the input been between 0 and 1, we would have said the graph was stretched horizontally.

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, . 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 and  due to the limited amount of information given for to share during the discussion.

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).