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.

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.

Students who interpret the “” in Andre's rule to mean "negative " (rather than "the opposite of ") may be unsure how to use that information. Ask students to evaluate the function for specific values of and to write down each step. For instance, when is -2, is , which is 2.

Select previously identified students to display their graph of function . Ask students who plotted only the ordered pairs in the table whether the pairs that are not in the table, if plotted, would also fall on the same two lines. Emphasize that can be shown with two lines.

Next, ask students to share their response to the last question. Students may find Elena's rule easier to explain because of their work with absolute value in recent activities. They may struggle to explain Andre's rule.

We can reason about Andre's equation a couple of ways:

• By thinking about rules: The equation is that of a piecewise function because different rules are applied to different parts of the domain to give “the distance from 0” as the output:

  • When the input is positive or 0, its distance from 0 is that same number, .
  • When the input is negative, its distance from 0 is the opposite of the number, .

• By using the graph: The two halves of the graph are lines with different slopes.

  • If we cover up the left side of the vertical axis and see only positive values of , we see a line with a slope of 1 that represents .
  • If we cover up the right side and see only negative values of , we see a line with a slope of -1 that represents

Explain to students that:

• Function is the absolute value function. It gives the distance of an input value from a certain value, the origin in this case.
• The graph of function is a V shape with the two lines of points meeting at , which is the minimum of the graph. We call this point the vertex of the graph. It is the point where the graph changes direction.
• The absolute value function is also a piecewise function because different rules are applied to different parts of the domain to get the outputs.

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.

Invite previously selected students to share their reasoning about the position of the graphs with different parameters. Sequence the discussion of the approaches by the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions, such as:

• “In a guessing game, if the target number is 2, would the absolute guessing error be calculated using or ?” (It would be .)
• “If the absolute guessing error function for the target number of 2 is graphed, where is the vertex? Use your strategy from this activity to explain.” (The vertex is at . This is because the graph will look like , but shifted to the right 2.)
• “How does the function and graph change if the target number is -2?” (It would be , and the graph would have a similar shape, but with the vertex at .)

If time permits and if not already mentioned by students, remind students that each function can be seen as a piecewise function with two parts, each part being a linear function.

Graphing the two linear functions gives two lines that intersect on the horizontal axis. For , the two lines meet at . For , they meet at .

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.

Select previously identified students to share their strategies for making a match. If students use the strategies listed in the Activity Narrative, order their presentation as shown.

If time permits, ask students to use a strategy that they find effective to describe the graphs of and .