Untitled Document

Lesson 1

Title: Designing with Parabolas

Author: Shelly Rogers

Grade Level: Pre-Calculus

Concepts: Students will gain a better understanding of how to manipulate the basic parabolic function in order to create parabolas that can be used to recreate specific images. Through this activity, students will be able to predict more easily the desired equation needed to represent a particular relationship, rather than having to simply “guess and check.” The students will also be able to see similarities between how different types of functions are manipulated.

Objectives:

Students will be able to:

1) discover the factors that influence translations and scaling of functions

2) predict the behavior and placement of functions

3) formulate equations of functions that best describe particular relationships

TEKS:

§111.35. Precalculus (One-Half to One Credit).

(b) Introduction.

(1) In Precalculus, students continue to build on the K-8, Algebra I, Algebra II, and Geometry foundations as they expand their understanding through other mathematical experiences. Students use symbolic reasoning and analytical methods to represent mathematical situations, to express generalizations, and to study mathematical concepts and the relationships among them. Students use functions, equations, and limits as useful tools for expressing generalizations and as means for analyzing and understanding a broad variety of mathematical relationships. Students also use functions as well as symbolic reasoning to represent and connect ideas in geometry, probability, statistics, trigonometry, and calculus and to model physical situations. Students use a variety of representations (concrete, numerical, algori\thmic, graphical), tools, and technology to model functions and equations and solve real-life problems.

(1) The student defines functions, describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, radical, exponential, logari\thmic, trigonometric, and piecewise-defined functions. The student is expected to:

(A) describe parent functions symbolically and graphically, including y = xⁿ, y = ln x, y = log_ax, y = , y = e^x, y = a^x, y = sin x, etc.;

(B) determine the domain and range of functions using graphs, tables, and symbols;

(D) recognize and use connections among significant points of a function (roots, maximum points, and minimum points), the graph of a function, and the symbolic representation of a function; and

(2) The student interprets the meaning of the symbolic representations of functions and operations on functions within a context. The student is expected to:

(A) apply basic transformations, including a • f(x), f(x) + d, f(x - c), f(b • x), |f(x)|, f(|x|), to the parent functions;

(3) The student uses functions and their properties to model and solve real-life problems. The student is expected to:

(A) use functions such as logari\thmic, exponential, trigonometric, polynomial, etc. to model real-life data;

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Materials List and Advanced Preparations:

· Computer (1/group)

· Graphmatica (already downloaded on computer)

· ExploreLearning Gizmo: Translating and Scaling Functions (already downloaded on computer)

· Printer accessibility and printing paper —OR— graphing paper

· Images predominately consisting of parabolas (different one for each group)

Engagement:

Teacher: * draws a parabola on the board.

* asks students if linear functions could be used to describe it.

Students: * eventually answer “Yes” and explain that if you use a lot of lines, with small domains, the curve could be approximated

Teacher: * asks students if this is efficient.

* asks if anyone knows something that may better and more easily/efficiently describe this curve

Students: * may or may not know the answer (parabola)

Exploration:

What the Teacher Will Do	What the Students Will Do	Ongoing/Formative Evaluation (Questions you will ask the students)*
Instruct groups to get a computer (unless they already have one at their table), turn it on, and open the Gizmo program	Students will get computer, turn it on, and open the program.
If students were unable to answer the last engagement question, the teacher asks which “parent” function most resembles the curve on the board.	The students answer parabola if they didn’t already.
Instruct students to select the parabola parent function and “play around” with the values of the function that can be changed.	Students explore parabolas using Gizmo
Teacher hands each group an image that predominately, if not fully, consists of parabolas.	Students continue exploring the Gizmo program or stop to observe the image that they were given.
Teacher gets the classes attention and tells the groups that they are to recreate their image using the Graphmatica program. The Gizmo program should simultaneously be used as a tool to help them determine how to create the image. The groups need to keep a log of all equations used to recreate their image and should know which equation corresponds to which function in their image. Teacher also tells the class how much time they have to work on this before she brings the groups back together for presentation and discussion.	Students open Graphmatica and begin their attempt to recreate the image. Students will either draw their image by hand, or print the image from Graphmatica.
Teacher walks around the classroom making sure the students understand their task, answering questions, and ensuring the students’ participation. (The teacher can and should do this by asking the students questions about their project.)	Students should be working as a group to recreate their image and making their equations log.
The teacher brings the groups back together as a class, so that they have enough time to present their images and discuss what they have learned.	The groups should have completed the recreation of their images. They should have either printed their newly created images or drawn them by hand. They should provide some key to explain which function corresponds to which equation in their log.

Explanation:

The groups present their images.
Teacher asks the students to compare two of their parabolas. “What is one way that one of the two parabolas’ equations is different from the other one’s equation, and how does this make their images different from each other? (should mention the effects of changing any of the 4 constants a, b, h, and k)
Discuss how they were able to manipulate the basic equation for a parabola in order to create parabolas that would best describe curves in the original image. (y = a(f(b(x-h))) + k, so we can manipulate a, b, h, k…we can also choose the domain of the functions.)

Elaboration:

Discuss how we’re still limited in our ability to create images if we only have linear and parabolic functions
Brainstorm other possible types of functions.

Evaluation:

The recreation of their image
The extent to which they participated
Their ability to explain the effects of changing certain aspects of a parabola’s equation
Short post-assessment

1.The graph of y = f(x − 3) is a _____ of the graph of y = f(x).

a. translation of 3 units to the right c. scale change of the output by a factor of -3

b. translation of 3 units up d. scale change of the input by a factor of 3

2.If the blue graph shown below is y = f(x), what is the red graph?

a. y = -5f(x) b. y = f(5x) c. y = f(x – 5) d. y = f(x) – 5

3.If the point (1, 3) is on the graph of y = f(x), then what is the corresponding point on the graph of y = −2f(x)?

a. (-2, -6) b. (1, -6) c. (-2, 3) d. (1, 3)

4.The point (4, 16) is on the graph of f(x) = x². What is the value of b if the point (1, 16) is on the graph of f(bx)?

a. 4 b. 0.5 c. 1 d. 2.5

Safety:

Students must conduct themselves in an orderly manner and respect the equipment.
No running in the classroom regardless; however, students should be especially aware of the computer chords.

What the Teacher Will Do

What the Students Will Do