Some students may struggle to plot the data without explicitly computing the absolute guessing errors first or creating a table of values. Encourage them to take those intermediate steps if they are helpful.

Select a student to display the completed scatter plot. Ask students:

• "How is the scatter plot for this data like the scatter plot for the absolute guessing errors from an earlier lesson?" (The points still form a V shape. There are still no negative -values.)
• "How are they different?" (The graphs are shifted horizontally toward the vertical axis. The two parts of the V now meet at .)

To help students see that the "actual average temperature" is like the "actual number of items in a jar" they saw earlier, highlight that:

• Earlier, we saw that when the actual number of items is , the absolute guessing error is "the distance of guess from ," which can be expressed as "the absolute value of (guess - )," or . 
• Here, likewise, when the actual average temperature is and the guess is , the absolute guessing error, , is "the distance of from ," which can be written as .
• When the actual temperature in Toronto is 0 degrees Celsius,  is "the distance of from 0," which can be written as  , or simply  .

This activity formally introduces students to the absolute value function as the function that takes an input value and gives its distance from the origin as the output. Students see two different ways to represent this function algebraically: using the absolute value notation and using the cases notation:

The equation in cases notation focuses on what has to happen algebraically to an input value to give its distance from the origin. Different rules apply to different intervals of the domain.

As students work, look for those who plot points for the graph of function and those who sketch two lines. Ask students who sketch different types of graphs to share during discussion.

In this activity, students expand their awareness of absolute value functions by analyzing the graphs and equations of several absolute value functions. The graphs have the same V shape but not the same vertex. The expressions that define the functions have a constant term added to or subtracted from the input, , or from the absolute value of the input, . Students observe how each constant term affects the graph and consider possible explanations.

Dynamic graphing technology can be very useful for observing how the parameters of a function are related to the features of its graph and can help students generalize their observations. Consider giving students individual access to dynamic graphing technology and then giving them time to explain the behaviors of the graphs. If students don't have individual access, projecting the applet in the digital Launch would be helpful.

Although digital tools can help students notice the connection between the parameters and the graph easily, it is still important for students to understand why they behave in that way.

Monitor for students who use these different strategies:

• Find some input-output pairs for each function and verify that the graph contains those pairs of values.
• Interpret the expressions in terms of finding an absolute error of a guess.
• Use the position of the number added or subtracted to reason about how the inputs and outputs are affected in the graph.

Plan to have students present in this order to support moving students from plotting points to thinking of functions as an object that can be translated.

In the digital version of the activity, students use an applet to make dynamic graphs and see how adjusting parameters translates functions. The applet allows students to see how the different values affect the graph in real time. Use the digital version if available so that students can more easily see the connections between the values added to different parts of the function and the result for the graph.

This optional activity gives students a chance to test the observations or apply the generalizations they made in the previous activity. They match equations and graphs of other absolute value functions, then sketch the graph of an equation without a match.

Students also encounter new cases in which a constant term is added or subtracted both before and after absolute value is applied to the input, resulting in a graph that shifts both vertically and horizontally relative to the graph of .

In making the matches (and before using technology to check their graphs), students are likely to reason in different ways and rely on structure to varying degrees (MP7). For instance, students may:

• Evaluate each function at different input values and then match the input-output pairs to the coordinates on the graphs.
• Start with the graph of and then shift sideways for addition or subtraction inside the absolute value symbols, and then shift the graph vertically for addition or subtraction after absolute value is applied.
• Think about the -value that would produce the smallest possible output and relate that ordered pair to the vertex of the graph.

Monitor for students using different strategies, and invite them to share during discussion.

Students will have opportunities to explore transformations of functions and their graphs in a later unit and more formally in a future course.