standard_description: Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
problem_activity: Warm-up
In this warm-up, students are prompted to compare function values. To do so, they need to interpret statements in function notation and connect their interpretations to the graph of the function.
Previously, students recognized that if a point with coordinates
\((2,20)\)
is on the graph of a function, the 2 is an input and the 20 is a corresponding output. Here, they begin to see the second value in a coordinate pair more abstractly, as
\(f(2)\)
when the input is 2, or as
\(f(4)\)
when the input is 4.
Launch
Give students 1–2 minutes of quiet think time. Follow with a whole-class discussion.
Some students might be unfamiliar with drones. If needed, give a brief explanation of what they are.
Student Facing
[IMG]
Here is a graph that represents function
\(f\)
, which gives the height of a drone, in meters,
\(t\)
seconds after it leaves the ground.
[IMG]
Decide which function value is greater.
\(f(0)\)
or
\(f(4)\)
\(f(2)\)
or
\(f(5)\)
\(f(3)\)
or
\(f(7)\)
\(f(t)\)
or
\(f(t+1)\)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Invite students to share how they compared each pair of outputs. If not mentioned in students’ explanations, point out that for each pair, what we are comparing are the vertical values of the points on the graph at different horizontal values. Even though the statements don’t tell us the values of, say
\(f(3)\)
and
\(f(7)\)
, and the vertical axis shows no scale, we can tell from the graph that the function has a greater value when
\(t\)
is 3 than when
\(t\)
is 7.
Tell students that the coordinates
\((0, f(0))\)
represent the starting point of the drone. Ask students, “What are the coordinates of the points when the drone starts leveling off? When it starts to descend? When it lands?” (
\((2, f(2)), (5,f(5))\)
, and
\((7,f(7))\)
, respectively.)
Highlight that the coordinates of each point on a graph of a function are
\((x, f(x))\)
.
Discuss with students how they compared
\(f(t)\)
and
\(f(t+1)\)
(in the last question). Make sure students see that the
\(f(t+1)\)
could be less than, equal to, or greater than
\(f(t)\)
, depending on the value of
\(t\)
.
If
\(t\)
is 2, then
\(t+1\)
is 3. From the graph, we can see that
\(f(2)\)
is equal to
\(f(3)\)
because the graph has the same vertical value when
\(t\)
is 2 and 3.
If
\(t\)
is 5, then
\(t+1\)
is 6. From the graph, we can see that
\(f(5)\)
is greater than
\(f(6)\)
because the graph has a greater vertical value when
\(t\)
is 5 than when
\(t\)
is 6.


aligns: no