Topicmefc08c1b2a40ab62_1528449000663_0Topic

Determining the monotonicity of the function

Levelmefc08c1b2a40ab62_1528449084556_0Level

Third

Core curriculummefc08c1b2a40ab62_1528449076687_0Core curriculum

V. Functions. The student:

4) reads from the plot of the functionplot of the functionplot of the function: the domain, the range, roots, monotonic intervals, intervals in which the function takes values not greater (not smaller) or smaller (not greater) than a given number, greatest and smallest values of the functionvalues of the functionvalues of the function (if they exist) in the closed interval and arguments for which the function takes greatest and smallest values.

Timingmefc08c1b2a40ab62_1528449068082_0Timing

45 minutes

General objectivemefc08c1b2a40ab62_1528449523725_0General objective

Using mathematical objects, interpreting mathematical concepts.

Specific objectivesmefc08c1b2a40ab62_1528449552113_0Specific objectives

1. Determining the monotonicity of the function.

2. Identifying and drawing plots of monotonic functions.

3. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesmefc08c1b2a40ab62_1528450430307_0Learning outcomes

The student:

- determines the monotonicity of the function,

- identifies and draws plots of monotonic functions.

Methodsmefc08c1b2a40ab62_1528449534267_0Methods

1. Discussion.

2. Situational analysis.

Forms of workmefc08c1b2a40ab62_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmefc08c1b2a40ab62_1528450127855_0Introduction

The teacher introduces the subject of the lesson - determining the monotonicity of the function and identifying and drawing plots of monotonic functions.

At home students loo for information about the monotonicity of functions and what functions are called monotonic.

Proceduremefc08c1b2a40ab62_1528446435040_0Procedure

Task
Students work individually, using computers. Their task is to observe how the values of the functionvalues of the functionvalues of the function change while the change of arguments.

[Geogebra applet]

Students define the decreasing functiondecreasing functiondecreasing function.

Definition of the decreasing function:
A function is called decreasing if the values of the function decrease as the argument increase.mefc08c1b2a40ab62_1527752256679_0A function is called decreasing if the values of the function decrease as the argument increase.

Discussion – what function can be called increasing or constant? Students analyse the problem based on their own examples of plots of functions.

Definition of the increasing functionincreasing functionincreasing function:
A function is called increasing if the values of the functionvalues of the functionvalues of the function increase as the arguments increase.

Definition of the constant functionconstant functionconstant function:
A constant function takes the same value independent of the argument.

Discussion – students order information about the monotonic functionmonotonic functionmonotonic function they prepared at home.

Conclusion:

A monotonic function is a function that keeps some kind of order. It means the function is increasing, non‑decreasing (increasing or constant), decreasing or non‑increasing (decreasing or constant).mefc08c1b2a40ab62_1527752263647_0A monotonic function is a function that keeps some kind of order. It means the function is increasing, non‑decreasing (increasing or constant), decreasing or non‑increasing (decreasing or constant).

Task
Determine the monotonicity of function f presented on the plot.

[Illustration 1]

Task
Draw a plot of a constant functionconstant functionconstant function whose domain is the set X = { -2, -1, 0, 2, 4}.
How many numbers are included in the range of this function?

Task
In the picture there is a plot of the functionplot of the functionplot of the function.

[Illustration 2]

a) Give the arguments for which the function is constant.
b) Give the maximal length of the interval in which the function is increasing.
c) Is the function monotonic?

Task
Based on the tabletabletable, determine if the function is decreasing, increasing or constant. Draw w plot of the function f.

[Table 1]

An extra task:
Draw a plot of non‑decreasing function that is defined for x $\in$ {1, 2, 3, 4, 5, 6}.

Lesson summarymefc08c1b2a40ab62_1528450119332_0Lesson summary

Students do the revision exercises.

Then together they sum‑up the classes, by formulating the conclusions to memorise.

Definition of the decreasing functiondecreasing functiondecreasing function:
A function is called decreasing if the values of the functionvalues of the functionvalues of the function decrease as the argument increase.

Definition of the increasing functionincreasing functionincreasing function:
A function is called increasing if the values of the function increase as the arguments increase.

Definition of the constant functionconstant functionconstant function:
A constant function takes the same value independent of the argument.

Selected words and expressions used in the lesson plan

increasing functionincreasing functionincreasing function

decreasing functiondecreasing functiondecreasing function

constant functionconstant functionconstant function

monotonic functionmonotonic functionmonotonic function

plot of the functionplot of the functionplot of the function

tabletabletable

integersintegersintegers

increasing argumentincreasing argumentincreasing argument

values of the functionvalues of the functionvalues of the function