standard_description: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
problem_activity: Activity
This activity focuses on a specific aspect of the previous activity: combining two functions to make a new function. Here, students sketch the graph of the function defined as the sum of two original functions. To help students make generalizations about the process, in each question one of the starting functions is always the same while the other is either constant, linear, or quadratic (MP8).
Monitor for students who approach the activity in different ways based on how precise they make their sketch, such as:
making a rough sketch by observation
plotting a few points by summing together the
\(y\)
-values of the two functions and then connecting them with a curve
measuring the distance of each function from the
\(x\)
-axis in a few locations and then combining those heights to determine the
\(y\)
-value of the new function at specific values of
\(x\)
making a table of values for
\(E\)
and the other function, as in the previous activity, in order to find the sums before making a sketch of the new function
Also look out for possible incorrect approaches due to students not considering the numerical values of the functions and what it means to add, such as by:
sketching a curve that is halfway between the two given graphs
letting the horizontal asymptote dictate end behavior on the right regardless of the value of the other function
Ask students who tried either of these last two approaches if they would be willing to share during the discussion. If possible, identify students who first did a sketch incorrectly and then revised their work to share at the end of the discussion.
Launch
Arrange students in groups of 2–3. Tell students that in this activity they are going to sketch graphs of functions that are the sum of two other functions. Ask students to first complete the sketch of
\(S\)
on their own and then check their work with their group before continuing on to the other graphs. Depending on time available, assign each group 3–5 of the graphs to complete.
Since this activity was designed to be completed without technology, ask students to put away any graphing devices.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, students can use different color highlighters for each graph. Students can identify and label points on each graph. Remind students that they are sketching the sum of the two functions.
Supports accessibility for: Visual-spatial processing
Student Facing
Here are the graphs of two functions,
\(E\)
and
\(L\)
. Define a new function
\(S\)
by adding
\(E\)
and
\(L\)
, so
\(S(x) = E(x) + L(x)\)
. On the same axes, sketch what you think the graph of
\(S\)
looks like.
[IMG]
Sketch the graph of the sum of
\(E\)
and each of the following functions.
[IMG]
[IMG]
[IMG]
[IMG]
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
Here are the graphs of two functions,
\(U\)
and
\(V\)
. Define a new function
\(W\)
by multiplying
\(U\)
and
\(V\)
, so
\(W(x) = U(x) V(x)\)
. On the same axes, sketch what you think the graph of
\(W\)
looks like.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The purpose of this discussion is for students to share how they made their sketches, focusing on how the numerical value of the new function is defined by the sum of the numerical values of the two original functions. The discussion begins with students sharing techniques that worked for them and then focuses on techniques that did not work and why, calling back to the strategies that did work in order to make sense of what went wrong.
Invite previously identified students who graphed correctly to share their process in the order listed in the Activity Narrative. Display student graphs and any accompanying work for all to see during the explanation. Since these were only sketches, students did not have to make a detailed table, but using a table illustrates clearly that adding two functions together means adding together their output values.
Next, invite previously identified students who took an incorrect approach to share in the order listed in the Activity Narrative. Where possible, identify what function the student did graph. For example, if a student sketched the curve halfway between the two starting graphs, the function they graphed is the average of the two starting graphs instead of the sum. If a student’s sketch appears to have a horizontal asymptote at
\(y=0\)
on the right side, it's possible they were thinking more of a function defined by the product of the two original functions.
If time allows, ask students to choose one of the questions and sketch the graph of a function defined by the
difference
between
\(E\)
and the other function.


aligns: no