Topicmee386c0af7b04373_1528449000663_0Topic

Plots of unusual functions

Levelmee386c0af7b04373_1528449084556_0Level

Third

Core curriculummee386c0af7b04373_1528449076687_0Core curriculum

V. Function. Basic level. The student:

4) reads from the graph of the functionfunctionfunction: the domain, the rangerangerange, 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 function (if they exist) in the closed interval and arguments for which the functionfunctionfunction takes greatest and smallest values.

Timingmee386c0af7b04373_1528449068082_0Timing

45 minutes

General objectivemee386c0af7b04373_1528449523725_0General objective

Interpreting and operating information presented in the text, both mathematical and popular science texts, as well as in the form of graphs, diagrams, tables.

Specific objectivesmee386c0af7b04373_1528449552113_0Specific objectives

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

2. Learning properties of some unusual functions.

3. Drawing plots of unusual functions.

Learning outcomesmee386c0af7b04373_1528450430307_0Learning outcomes

The student:

- learns properties of some unusual functions,

- draws plots of unusual functions.

Methodsmee386c0af7b04373_1528449534267_0Methods

1. Talking cards.

2. Situational analysis.

Forms of workmee386c0af7b04373_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmee386c0af7b04373_1528450127855_0Introduction

Students revise information about functions using the method of ‘talking cards’.

Each student gets a piece of paper where they write their associations with functions. Cards are collected by one of the students who reads their content.

The teacher verifies answers and clarifies doubts.

Proceduremee386c0af7b04373_1528446435040_0Procedure

The teacher introduces the subject of the lesson – learning properties of some unusual functions.

Students look for information about the phrase ‘an integer part of xinteger part of xinteger part of x’ in available knowledge sources. They formulate a definition.

Definition
[The integer part of x is the greatest integer number not greater than x. It is denoted as [x].]\dymek‑ref={mee386c0af7b04373_1527752263647_0}

Task
Using the definition, students calculate:

a) [7,54],

b) [- 2,43],

c) [0,235].

TaskStudents work in groups and analyse the Interactive illustration 1. They read properties of the functionfunctionfunction f(x) = [x] from the plotplotplot. They write them as conclusions.

[Interactive illustration 1]

Conclusion:

[1. The domain of the function f(x) = [x] is the set of real numbers.2. The range is the set of integer numbers.3. It is a non‑decreasing function.4. It is not an odd or even function.]\dymek‑ref={mee386c0af7b04373_1527752256679_0}

Task
Students work in groups and draw plots of functions:

a) f(x) = [2x],

b) f(x) = [ - 3x + 2].

Students work on their own and look for information about the functionfunctionfunction f(x) = sgn(x) in available knowledge sources. They give definitions of this function.

Definition
Signum (lat. Signum „sign”) – the function of the real variable, defined as follows:

.

Task
Students work in groups and analyse the Interactive illustration 2. They read properties of the functionfunctionfunction f(x) = sgn(x) from the plotplotplot. They write them as conclusions.

[Interactive illustration 2]

Conclusion

Properties of functionfunctionfunction f(x) = sgn(x):

1. The domain of the functiondomain of the functiondomain of the function is the set of real numbersreal numbersreal numbers.

2. The rangerangerange is {- 1, 0, 1}.

3. The function has one rootrootroot x = 0.

4. The functionfunctionfunction is odd, non‑decreasing, non‑injective.

Students work in groups and do exercises.

Task
Draw the plot of the functionplot of the functionplot of the function f and describe its properties:

a) $f\left(x\right)=\mathrm{sgn}\left(x\right)$,  

b) $f\left(x\right)=\mathrm{sgn}\left(x\right)$, 

c) $f\left(x\right)=-\left(\mathrm{sgn}\right(x\left)\right)$, $x\in <-\infty ,5>$

Task
Draw the plot of the functionplot of the functionplot of the function f, given its roots and rangerangerange:

a) $f\left(x\right)=\left[x\right]$, $x\in ⟨-4,6⟩$

b) $f\left(x\right)=\left[x\right]$, $x\in ⟨-2,8)$

After having completed exercises, student present obtained results and grade their own work.

The teacher verifies answers and clarifies doubts.

An extra task
Draw the plot of the functionplot of the functionplot of the function $f\left(x\right)=x-\left[x\right],$$x\in <-7,6>$ and describe its properties.

Lesson summarymee386c0af7b04373_1528450119332_0Lesson summary

Students do the revision exercises.

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

- The integer part of x is the greatest integer number not greater than x. It is denoted as [x].
- Signum (lat. Signum „sign”) – the function of the real variable, defined as follows:

.

Selected words and expressions used in the lesson plan

domain of the functiondomain of the functiondomain of the function

functionfunctionfunction

integer part of xinteger part of xinteger part of x

plotplotplot

plot of the functionplot of the functionplot of the function

rangerangerange

range of the functionrange of the functionrange of the function

real numbersreal numbersreal numbers

rootrootroot

signum functionsignum functionsignum function