Lesson 11

This warm-up prompts students to carefully analyze and compare the properties of four graphs. Each graph represents a function, but no labels or scales are shown on the coordinate axes, so students need to look for and make use of the structure of the graphs in determining how each one is like or unlike the others (MP7).

In making comparisons, students have a reason to use language precisely (MP6), especially mathematical terms that describe features of graphs or properties of functions.

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. Encourage students to use relevant mathematical vocabulary in their explanations, and ask students to explain the meaning of any terminology that they do use, such as "intercepts," "minimum," or "linear functions."

After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

By now students are well aware that certain features of a graph have special significance in that they tell us something about the quantities or relationships in the situation. Tell students that features of graphs can also help us understand the domain and range of a function. We will explore this in the lesson.

In this activity, students are given the same four graphs they saw in the warm-up and four descriptions of functions and are asked to match them. All of the functions share the same context. Students then use these features to reason about likely domain and range of each function.

To make the matches, students analyze and interpret features of the graphs, looking for and making use of structure in the situation and in the graphs (MP7). Here are some possible ways students may reason about each match:

• The swing goes up and down while the child is swinging, so D could be a graph for function \(h\).
• The time left on the swing decreases as the time on the swing increases, so A is a possible graph for function \(r\).
• The distance of the child from the top beam of the swing doesn't change as long as the child is on the swing, so C is a possible graph for function \(d\).
• The total number of times the swing is pushed must be a counting number and cannot be fractional. Graph B has multiple pieces and each one could represent the total number of push for certain intervals of time, so B is a possible graph of \(s\).

Students reason quantitatively and abstractly as they connect verbal and graphical representations of functions and as they think about the domain and range of each function (MP2).

Ask students to imagine a child getting on a swing, swinging for 30 seconds, and then getting off the swing. Explain that they will look at four functions that can be found in this situation. Their job is to match verbal descriptions and graphs that define the same functions, and then to think about reasonable domain and range for each function. Tell students they will need additional information for the last question.

Arrange students in groups of 2. Give students a few minutes of quiet time to think about the first two questions, and then time to discuss their thinking with their partner. Follow with a whole-class discussion.

Invite students to share their matching decisions and explanations as to how they know each pair of representations belong together. Make sure that students can offer an explanation for each match, including for their last pair (other than because the description and the graph are the only pair left). See some possible explanations in the Activity Narrative.

Next, ask students to share the points that they think would be helpful for determining the domain and range of each function. If students gesture to the intercepts, a maximum, or a minimum on a graph but do not use those terms to refer to points, ask them to use mathematical terms to clarify what they mean.

Here is the information students will need for the last question. Display it for all to see, or provide it as requested. If the requested information is not shown or cannot be reasoned from what is available, ask students to request a different piece of information.

• The child is given 30 seconds on the swing.
• While the child is on the swing, an adult pushes the swing a total of 5 times.
• The swing is 1.5 feet (18 inches) above ground.
• The chains that hold the seat and suspend it from the top beam are 7 feet long.
• The highest point that the child swings up to is 4 feet above the ground.

If time is limited, ask each partner to choose two functions (different than their partner's) and write the domain and range only for those functions.

Some students may struggle to match the descriptions and the graphs because they confuse the independent and dependent variables and think that, in each situation, time is represented by the vertical axis. Encourage them to re-read the activity statement, clarify the input and output in each situation, label the horizontal graph with the input, and then try interpreting the graphs again.

Select students to share the domain and range for each function and their reasoning. Record and display their responses for all to see.

One key point to highlight is that the range of a function could be a single value (say 7, as shown in graph C), a bunch of isolated values (say, only some whole numbers, as shown in graph B), all values in an interval (say, all values from 1.5 to 4, as shown in graph D, or all values between 0 and 30, as in graph A), or a combination of these.

The domain of a function may also be limited in similar ways. Tell students that, in upcoming lessons, we will look at functions in which the rules relating their input and output are a bit more complex, so their domain and range are also a bit more so.

In this activity, students continue to interpret a graph of a function in terms of a situation and relate the features of the graph to the domain and range of the function. The context is a familiar one, allowing students to focus their reasoning on domain and range.

Unlike in the activity about a child on a swing, the graph includes a scale on each axis and the coordinate pairs of some points, allowing students to identify the range more definitively. On the other hand, the graph is a partial representation of the function, as it does not show what happens after a dropped ball hits the ground the fifth time. When describing the domain, students need to attend to what is reasonable in this situation (that is, noting that the ball likely does not just stop after the fifth bounce).

Focus the discussion on how students reasoned about the domain and the range for the function. Highlight explanations that account for what is realistic in the context.

• The domain tells us all the possible amounts of time that passed since the moment the tennis ball was dropped until it stopped rolling.
• The range includes all the possible heights of the tennis ball from the time it was dropped until the time it stopped rolling.

Ask students if there are points on the graph whose coordinates are particularly useful for identifying the domain and range of the function. (The heights of the bounces? The points where the ball hits the ground? Points between the two? Others?)

Emphasize that, just like in the activity about the swing, some points and features on a graph can give us more information than others about possible input-output values of a function.