Key visualizations: Precalculus

Modeling is a key idea throughout the high school curriculum, and using data and problem situations motivates the study of new functions. In Precalculus, students model a variety of problem situations with functions. They expand their library of function models and deeply analyze their characteristics, behavior, and key features.

Connections to the TEKS []
This animation addresses the following Precalculus TEKS:

(2) Functions. The student uses process standards in mathematics to explore, describe, and analyze the attributes of functions. The student makes connections between multiple representations of functions and algebraically constructs new functions. The student analyzes and uses functions to model real-world problems. The student is expected to:
(G) graph functions, including exponential, logarithmic, sine, cosine, rational, polynomial, and power functions and their transformations, including af(x), f(x) + d, f(xc), f(bx) for specific values of a, b, c, and d, in mathematical and real-world problems;
(I) determine and analyze the key features of exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions such as domain, range, symmetry, relative maximum, relative minimum, zeros, asymptotes, ad intervals over which the function is increasing and decreasing;
(N) analyze situations modeled by functions, including exponential, logarithmic, rational, polynomial, and power functions, to solve real-world problems;


About this animation []

In this animation, students consider the relationship between the volume of air in a person’s lungs over time as they breathe. After considering the cyclical nature of the graph (foreshadowing the later study of periodic functions), students focus on a single cycle of the graph, analyze its behavior, and describe its key features. Students notice that the shape of the function is different from the functions they have studied previously. This motivates the need for general polynomial functions. Students use technology to experiment with parameters on a general cubic function to fit a function rule to the graph. Students can connect some of the parameters to vertical shifts and vertical stretches or compressions, but other parameters have a very different effect on the graph.

Students must persist as they explore parameters that affect the graph model in a different way from parameters they have studied in the past.


Olivia is resting. She completes one respiratory cycle (inhale and exhale) every 5 seconds. How could you use a graph to represent the volume of air in her lungs over time as she rests?
As Olivia inhales, the volume of air in her lungs increases. As she exhales, the volume of air in her lungs decreases. The cycle repeats every 5 seconds.
TBD
As an athlete, Olivia has to train every part of her body, including her lungs. The act of breathing must be very deliberate in order to maximize the oxygen that will be available for her cells to use. Olivia has been working hard to control her breathing during workouts.
As Olivia releases from the crunch, she takes air into her lungs. As she exerts the effort for the next crunch, she blows the used air back out. Can you picture a graph that would model the volume of air in Olivia's lungs as she does crunches? How will this graph compare to the one used to model the volume of air in a resting person's lungs?
TBD
Depending on how Olivia paces her breaths, the graph might look something like this.
How well do the functions you have studied in the past model Olivia's lung capacity when she is doing a single crunch?
Perhaps a new function is needed.
During the first second, air volume increases rapidly. Then volume increases more slowly until the lungs are filled.
Do you see the slight change in the way the graph bends between 0 and 3 seconds?
TBD
Set the sliders to try to find a cubic function that models the lung capacity problem.
You found the correct value for a0. Use the sliders to modify the other coefficients and find a good fit.
You found the correct value for a1. Use the sliders to modify the other coefficients and find a good fit.
You found the correct value for a2. Use the sliders to modify the other coefficients and find a good fit.
You found the correct value for a3. Use the sliders to modify the other coefficients and find a good fit.
You found a possible model. The graph of this cubic seems to fit the lung capacity graph fairly well.
You found the correct values for a0 and a1. Use the sliders to modify the other coefficients and find a good fit.
You found the correct values for a0 and a2. Use the sliders to modify the other coefficients and find a good fit.
You found the correct values for a0 and a3. Use the sliders to modify the other coefficients and find a good fit.
You found the correct values for a1 and a2. Use the sliders to modify the other coefficients and find a good fit.
You found the correct values for a1 and a3. Use the sliders to modify the other coefficients and find a good fit.
You found the correct values for a2 and a3. Use the sliders to modify the other coefficients and find a good fit.
You found the correct values for a0, a1 and a2. Use the sliders to modify the other coefficients and find a good fit.
You found the correct values for a0, a1 and a3. Use the sliders to modify the other coefficients and find a good fit.
You found the correct values for a0, a2 and a3. Use the sliders to modify the other coefficients and find a good fit.
You found the correct values for a1, a2 and a3. Use the sliders to modify the other coefficients and find a good fit.