standard_description: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
problem_activity: Activity
Continuing the thinking from the warm-up, students now compare three functions whose graphs are vertical translations from one another and generalize their observations of the graphs and data tables into equations where one function is defined in terms of another.
Monitor for students making connections between representations as they write an equation for
\(g(x)\)
in terms of
\(h(x)\)
. For example, students may:
compare the graphs of
\(g\)
and
\(h\)
, perhaps sketching them on the same axes and then measuring the vertical distance between them
calculate the difference between the outputs from the table
combine the equation they wrote for
\(g(x)\)
in terms of
\(f(x)\)
and
\(h(x)=f(x)-2.5\)
to rewrite
\(g(x)\)
in terms of
\(h(x)\)
Launch
Representation: Internalize Comprehension.
Chunk this task into more manageable parts. Check in with students after the first 2–3 minutes of work time to look for students who notice the constant difference between the outputs of
\(f(x)\)
and
\(g(x)\)
. Check to make sure that students are filling out the
\(g(x)\)
column of the table correctly.
Supports accessibility for: Conceptual processing; Organization
Student Facing
The graph of function
\(g\)
is a vertical translation of the graph of polynomial
\(f\)
.
[IMG]
[IMG]
Complete the
\(g(x)\)
column of the table.
If
\(f(0) = \text-0.86\)
, what is
\(g(0)\)
? Explain how you know.
Write an equation for
\(g(x)\)
in terms of
\(f(x)\)
for any input
\(x\)
.
The function
\(h\)
can be written in terms of
\(f\)
as
\(h(x)=f(x)-2.5\)
. Complete the
\(h(x)\)
column of the table.
| \(x\) | \(f(x)\) | \(g(x)\) | \(h(x)=f(x)-2.5\) |
|-------|----------|----------|-------------------|
|  -4   |    0     |          |                   |
|  -3   |   -5.8   |          |                   |
| -0.7  |    0     |          |                   |
|  1.2  |   -3.3   |          |                   |
|   2   |    0     |          |                   |
Sketch the graph of function
\(h\)
.
[IMG]
Write an equation for
\(g(x)\)
in terms of
\(h(x)\)
for any input
\(x\)
.
Student Response
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Activity Synthesis
A key takeaway from this discussion is that students understand that a vertical translation from one graph to another corresponds to constant differences between the outputs of the functions for the same input. We can then generalize these point by point comparisons using function notation, where adding or subtracting from the output of the function resulting in a vertical translation of its graph.
Select previously identified students to share the connections they made between the representations in the order listed in the Activity Narrative. If time allows and no students used the equations for
\(g(x)\)
and
\(h(x)\)
in terms of
\(f(x)\)
to write a new equation for
\(g(x)\)
in terms of
\(h(x)\)
, invite them to do so now.


aligns: no