Instructional Unit: Power Functions

By: Ashley Jones

This particular instructional unit is intended for first year Algebra students, and aligns with the MM1A1 standard. Many of the included exercises and activities are suggested to be used through *Geometer's Sketchpad, **Graphing Calculator*, and *Microsoft Word.*

Objectives:

MM1A1. Students will explore and interpret the characteristics of functions, using

graphs, tables, and simple algebraic techniques.

a. Represent functions using function notation.

b. Graph the basic functions f(x) = x^n, where n = 1 to 3, f(x) = sqrt(x), f(x) = |x|,

and f(x) = 1/x.

c. Graph transformations of basic functions including vertical shifts, stretches,

and shrinks, as well as reflections across the x- and y-axes.

d. Investigate and explain the characteristics of a function: domain, range,

zeros, intercepts, intervals of increase and decrease, maximum and minimum

values, and end behavior.

e. Relate to a given context the characteristics of a function, and use graphs and

tables to investigate its behavior.

f. Recognize sequences as functions with domains that are whole numbers.

g. Explore rates of change, comparing constant rates of change (i.e., slope)

versus variable rates of change. Compare rates of change of linear, quadratic,

square root, and other function families.

h. Determine graphically and algebraically whether a function has symmetry

and whether it is even, odd, or neither.

i. Understand that any equation in x can be interpreted as the equation

f(x) = g(x), and interpret the solutions of the equation as the x-value(s) of the

intersection point(s) of the graphs of y = f(x) and y = g(x).

Day 1:

Start the power functions unit with a review lesson on linear functions, since f(x) = x is also considered a power function. Use the GSP file, linear functions (pages 1 and 2), to start the lesson. Hand out the 'Gas Prices' worksheet for an extended application problem on the topic.

Day 2:

Introduce the basic functions from part b of the objectives standard. First start with the *Graphing Calculator *file (click on the blue box) which illustrates three basic power functions with n ranging from 1 to 3, . Then, begin class discussion based on the GSP file 'Power Functions.'

**Day 3:**

Continue with power functions by discussing the concepts of domain/range, vertical shifts, stretches, compressions, and reflections. Use the GSP file, power functions (pages 3 through 8) and appropriate Graphing Calculator files (vert. shifts, vert. stretches/compressions, and horiz. stretches/compressions), to lead the lesson for the day.

**Day 4:**

This day of the unit will be used to quiz the students over the material examined in the first three days of the instructional unit. This quiz will incorporate an applied linear function problem, graphing of different power functions, and descriptions of shifts/stretches/compressions/reflections for given power function graphs. The quiz should take students around 25 minutes, leaving the rest of the class period (about 20 to 25 more minutes) to write a journal entry about power functions. In this journal entry I expect students to discuss what things they have learned so far that interest them and/or things they did not previously know, things from this unit of material that they feel like they are confused about still, and things about power functions they hope to learn during the remaining part of the unit. The rubric for the journal entry is in the Word document 'Journal Rubric.'

**Day 5: **

Introduce quadratic functions including vertex form of a quadratic function, how to find the vertex, intervals of increasing and decreasing, and maximum/minimum values. Use the GSP file, quadratic functions, to lead this lesson with the class.

**Day 6**:

Provide students with the definition and understanding of what is a solution of an equation. Introduce and work on the quadratic function. The support material for this lesson can be found on the GSP file quadratic functions.

**Day 7:**

Students will spend this day of the unit using TI-nspire to find the power function that best fits the different 'activities.' Each group of four students will be able to use the speed attachments to determine and discover on their own the appropriate best fit functions. Each group will be given a marble to roll across the table, a yo-yo, and a bouncey ball.

**Day 8:**

Students will be given an activity to work on in pairs where they are able to model the area of a base of a box, as well as work on finding the equations of area of the base, and volume of the box. Have students attempt the activity and come up with solution ideas for the first 10 minutes of class. Also, have students work in pairs to discuss the possible solutions from the applied problem. Found worked out on GSP file: Open Top Box

**Day 9:**

Review for the unit test.

**Day 10:**

Test the students' knowledge over the unit of power functions. This will include an assortment of testing questions including extended problems requiring the students to apply their knowledge about the material.