options: Build new functions from existing functions.

Understand the concept of a function and use function notation.

Use functions to model relationships between quantities.

Build a function that models a relationship between two quantities.

Prove and apply trigonometric identities.

Define, evaluate, and compare functions.

Model periodic phenomena with trigonometric functions.

Interpret functions that arise in applications in terms of the context.

Analyze functions using different representations.

Interpret expressions for functions in terms of the situation they model.

Construct and compare linear and exponential models and solve problems.

Extend the domain of trigonometric functions using the unit circle.
problem_activity: Activity
The purpose of this activity is to review how to create a scatter plot using technology and come up with a function to model the data. This work is similar to the work in the associated Algebra 1 lesson, but includes more explicit instructions and steps toward analyzing data and choosing an appropriate model (MP4).
Launch
Display the table of data for all to see, and ask students whether they think a linear or exponential model would be a better fit for this data. After a minute of quiet think time, ask a few students to share their reasoning.
If needed, demonstrate using graphing technology to input the given data and create a scatter plot.
If the graphing technology platform you are using allows it, consider showing students how to create sliders. In Desmos, simply type
\(y=mx+c\)
in the expressions list, and then click “all” to add sliders for both
\(m\)
and
\(c\)
. You may need to adjust the maximum value of one or both sliders to be able to use them to create a good model for this data.
Student Facing
In the spring, an agave plant sends up a flower spike. Here are some data collected from an agave plant in a garden in Tucson, AZ, starting on April 2:
[IMG]
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "day"}, {"_attributes": {"scope": "col"}, "_value": "height in inches"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "0"}, {"_value": "17"}]}, {"td": [{"_value": "1"}, {"_value": "23"}]}, {"td": [{"_value": "2"}, {"_value": "29"}]}, {"td": [{"_value": "3"}, {"_value": "37"}]}, {"td": [{"_value": "4"}, {"_value": "45"}]}, {"td": [{"_value": "5"}, {"_value": "52"}]}, {"td": [{"_value": "6"}, {"_value": "62"}]}, {"td": [{"_value": "7"}, {"_value": "70"}]}, {"td": [{"_value": "8"}, {"_value": "80"}]}]}]}]}
Use graphing technology to create a scatter plot, using days as the first coordinate and height as the second coordinate.
Would a linear or exponential model be a better fit for this data?
Create a function that is a good model for the data. If you chose an exponential model, start with the equation
\(y=a \boldcdot b^x\)
and select values for
\(a\)
and
\(b\)
. If you chose a linear model, start with the equation
\(y=mx+c\)
, and select values for
\(m\)
and
\(c\)
.
Graph your equation on that same coordinate plane as your scatter plot. Adjust the numbers you used in the equation to improve your model.
Explain what each number in your equation means in this situation.
Use your model to predict the height of the flower spike on day 10.
Describe any limitations on the domain of the function modeling the data.
Student Response
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Activity Synthesis
Display the scatter plot using technology. Explore some of the different equations students used as a model. Discuss which model is the better fit, and what makes a model a good fit. Invite students to share their responses to the last three questions.


correct_option_index: 10