LESSON PLAN FOR INVERSE FUNCTIONS

 

Name:  Janie McMillin

 

Title of Lesson:   Inverse Functions

 

Description of Class: High school class, Pre-Calculus

 

 

TEKS Addressed: 

                           (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, algorithmic, graphical), tools, and technology to model functions and equations and solve real-life problems.

 (2)  As students do mathematics, they continually use problem-solving, language and communication, connections within and outside mathematics, and reasoning. Students also use multiple representations, applications and modeling, justification and proof, and computation in problem-solving contexts.

                               (c)  Knowledge and skills.

 (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, logarithmic, trigonometric, and piecewise-defined functions. The student is expected to:

              (A)  describe parent functions symbolically and graphically, including y = xn, y = ln x, y = loga x, y = , y = ex, y = ax, y = sin x, etc.;

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

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

              (B)  perform operations including composition on functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, and graphically; and

                 (C)  investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properties.

(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 logarithmic, exponential, trigonometric, polynomial, etc. to model real-life data;

 

 

I.   OVERVIEW               

The students will learn how to interpret and graph an inverse trig. Function and will also learn to solve for an equation with an inverse function.  Then the students will apply this knowledge to the construction of their sundial. 


II. PERFORMANCE OR LEARNER OUTCOMES

            Students will:

1)    recognize relationships and properties between functions and inverse functions

2)     be able to graph inverse functions

3)    determine the limitations on the domain and range of inverse trig. functions

4)    be capable of solving for an equation involving inverse. Trig.  functions

 

Five E Organization

 

 

ENGAGEMENT- Because the students have started working on their template for the sundial we will have an open discussion on how to make the markers on the base for hour line angles.  An example opening question would be Òhow to we know how far apart to put our angles?Ó  or Òwhere does the mark for noon or 6Õo clock go?Ó Then the students would be required to answer the questions in mathematical terms, therefore the students will have to consider which math concept would apply to this design aspect.  This will lead into our topic of the day.

 

EXPLORATION- We will have to review what an inverse function is, this can be done by having the students creating their own inverse functions. By the students creating their own functions to graph this may help pick out any misconceptions the students may have. Then the students will be presented with the sine function.  The students in groups will have to decide if the sine function is one-to-one and based on that if it has an inverse function.  Then the students will take a section of the  sine function from a specific range (specifically -p/2<y< p/2)   and will have to graph out the inverse function of sine, give the domain and range, and decide how the inverse function should be symbolized.  They will also do this with cosine.

 

EXPLANATION- After the students have graphed out the inverse functions then we will compare the graphs and correct any mistakes that have been made, other groups will give feedback on the graph that is being presented and then we will draw a graph for the class together on the board.  Then the teacher will explain the correct symbolism to represent inverse trigonometry functions,properties of an inverse function, and how to algebraically and graphically represent it .  Then we will also show some properties of the inverse functions.  We will use the graph to show why arcsin(sin x) = x and sin(arcsin x) = x and arccos(cos x) = x and cos(arccos x) = x.  

 

ELABORATION- The students will apply these properties to find values of inverse functions.  The students will be given examples (next page) and be able to work on them in groups and then each group will have to show one problem and explain the problem to the rest of the class.  Any common errors or misconceptions that the students are having will hopefully be covered by giving tricky but not difficult examples. 

 

EVALUATION- After the students fill comfortable with these types of equations then they will have to relate it back to their sundial project.  Given the formula to find the hour line angles                                                      X = arctan {sin ¿ * tan (h)}

where h is the hour angle, in degrees, given by:

 

h = (T24 - 12) * 15¡

 

and T24 is the time in 24-hour clock notation (hours after midnight) in decimal hours.

 

The students will make the calculations for their specific sundial.

 

Examples:

 

         Find the value of each expression:

 

1)  sin(cos -1 (1/2) )                                                 2) tan (sin-1(-1/2))

 

 

3)sec(cos-1(1/2))                                                     4) csc(tan-1 1)