Topicm7eec261b663fed3d_1528449000663_0Topic

The graph of functiongraph of functiongraph of function $f\left(x\right)=\frac{a}{x}$

Levelm7eec261b663fed3d_1528449084556_0Level

Third

Core curriculumm7eec261b663fed3d_1528449076687_0Core curriculum

V. Functions. The student:

13) uses functionfunctionfunction $f\left(x\right)=\frac{a}{x}$, including its graph, to describe and interpret concepts connected with inversely proportional values, also in practical task.

Timingm7eec261b663fed3d_1528449068082_0Timing

45 minutes

General objectivem7eec261b663fed3d_1528449523725_0General objective

Interpretation and application of information presented in a text, both mathematical and popular science, also in a form of graphs, diagrams and tables.

Specific objectivesm7eec261b663fed3d_1528449552113_0Specific objectives

1. Communication in English, developing mathematical, IT and basic scientific and technical competence, developing learning skills.

2. Drawing functionfunctionfunction $f\left(x\right)=\frac{a}{x}$.

3. Defining the properties of function $f\left(x\right)=\frac{a}{x}$ on the basis of its graph.

Learning outcomesm7eec261b663fed3d_1528450430307_0Learning outcomes

The student:

- draws function $f\left(x\right)=\frac{a}{x}$,

- defines the properties of functionfunctionfunction $f\left(x\right)=\frac{a}{x}$ on the basis of its graph.

Methodsm7eec261b663fed3d_1528449534267_0Methods

1. Brainstorming.

2. Situation analysis.

Forms of workm7eec261b663fed3d_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm7eec261b663fed3d_1528450127855_0Introduction

The students recollect the relationship between two inversely proportional magnitudes. They use the brainstorming to find examples of inversely proportional magnitudes in their surroundings.

They write their observations on posters and present them on the board.

Procedurem7eec261b663fed3d_1528446435040_0Procedure

The teacher informs the students that the aim of the class is getting to know the graph of functiongraph of functiongraph of function describing inverse proportionality. They also get to know the graph of function $f\left(x\right)=\frac{a}{x}$, defined for any number $x$ different from zero.

The students draw the graph of inversely proportional functionfunctionfunction.

Task
Using the chart draw the graph of functiongraph of functiongraph of function $f\left(x\right)=\frac{4}{x}$ for $x>0$.

[Table]

Check the correctness of your drawing.

[Illustration 1]

Answer the following questions:
- In which quadrant of the coordinate systemcoordinate systemcoordinate system is the graph located?
- How is the graph located with respect to the axes of the coordinate systemcoordinate systemcoordinate system?

Working in groups, the students use the applet to analyse the graphs of function $f\left(x\right)=\frac{a}{x}$ for $x\ne 0$ and define their values.

Task
Analyse the material presented in the applet. Discuss the properties of functionfunctionfunction $f\left(x\right)=\frac{a}{x}$ with respect to the sign of coefficient $a$.

[Geogebra applet]

The students should notice that:

- The graph of function $f\left(x\right)=\frac{a}{x}$ is a hyperbola, located in the first and the third quadrant of the coordinate system when $a>0$ (in the second and the fourth quadrant when $a<0$).
- The arms of the hyperbola are located symmetrically with respect to the origin of the coordinate system.
- Lines $x=0$ and $y=0$ are the asymptotes of the graph.
- The domain of the function is a set of real number different from zero.
- The set of values is the set of real number different from zero.
- If $a>0$ function f is decreasing in every interval $(-\infty ,0)$ and $(0,+\infty )$.
- If $a<0$ function f is increasing in every interval $(-\infty ,0)$ and $(0,+\infty )$.The function has no roots.m7eec261b663fed3d_1527752263647_0- The graph of function $f\left(x\right)=\frac{a}{x}$ is a hyperbola, located in the first and the third quadrant of the coordinate system when $a>0$ (in the second and the fourth quadrant when $a<0$).
- The arms of the hyperbola are located symmetrically with respect to the origin of the coordinate system.
- Lines $x=0$ and $y=0$ are the asymptotes of the graph.
- The domain of the function is a set of real number different from zero.
- The set of values is the set of real number different from zero.
- If $a>0$ function f is decreasing in every interval $(-\infty ,0)$ and $(0,+\infty )$.
- If $a<0$ function f is increasing in every interval $(-\infty ,0)$ and $(0,+\infty )$.The function has no roots.

The students use the information to solve the tasks.

Task
Sketch the graph of functiongraph of functiongraph of function $f\left(x\right)=\frac{-4}{x}$. Give its least value and the greatest value in interval .

Task
Point P ( - 1, 8) belongs to the graph of function $f\left(x\right)=\frac{a}{x}$. Find the value of coefficient a.
Answer: a = -8m7eec261b663fed3d_1527752256679_0Point P ( - 1, 8) belongs to the graph of function $f\left(x\right)=\frac{a}{x}$. Find the value of coefficient a.
Answer: a = -8

An extra task:
Solve the task in the graphic representation.
There is a 200 kilometers long railway between towns A and B. What time is needed to cover the distance by:
1. A handcar moving with the speed of 25 $\frac{\mathrm{km}}{\mathrm{h}}$.
2. A railway safety inspector cycling along the rails with the speed of 12,5 $\frac{\mathrm{km}}{\mathrm{h}}$.
3. A goods train moving with the speed of 40 $\frac{\mathrm{km}}{\mathrm{h}}$.
4. A Stopping train moving with the speed of 100 $\frac{\mathrm{km}}{\mathrm{h}}$.

Mark the points corresponding to the times and speeds of particular vehicles. What relationship do you notice? Sketch the shape of this relationship.

[Illustration 2]

Lesson summarym7eec261b663fed3d_1528450119332_0Lesson summary

Students do the consolidation tasks and summarize the class by formulating conclusions to memorize:

Conclusions:

- The graph of function $f\left(x\right)=\frac{a}{x}$ is a hyperbola, located in the first and the third quadrant of the coordinate system when $a>0$ (in the second and the fourth quadrant when $a<0$).
- The arms of the hyperbola are located symmetrically with respect to the origin of the coordinate system.
- Lines $x=0$ and $y=0$ are the asymptotes of the graph.
- The domain of the function is a set of real number different from zero.
- The set of values is the set of real number different from zero.
- If $a>0$ function f is decreasing in every interval $(-\infty ,0)$ and $(0,+\infty )$.
- If $a<0$ function f is increasing in every interval $(-\infty ,0)$ and $(0,+\infty )$.The function has no roots.m7eec261b663fed3d_1527752263647_0- The graph of function $f\left(x\right)=\frac{a}{x}$ is a hyperbola, located in the first and the third quadrant of the coordinate system when $a>0$ (in the second and the fourth quadrant when $a<0$).
- The arms of the hyperbola are located symmetrically with respect to the origin of the coordinate system.
- Lines $x=0$ and $y=0$ are the asymptotes of the graph.
- The domain of the function is a set of real number different from zero.
- The set of values is the set of real number different from zero.
- If $a>0$ function f is decreasing in every interval $(-\infty ,0)$ and $(0,+\infty )$.
- If $a<0$ function f is increasing in every interval $(-\infty ,0)$ and $(0,+\infty )$.The function has no roots.

Selected words and expressions used in the lesson plan

asymptotes of a hyperbolaasymptotes of a hyperbolaasymptotes of a hyperbola

coordinate systemcoordinate systemcoordinate system

domain of the functiondomain of the functiondomain of the function

functionfunctionfunction

graph of functiongraph of functiongraph of function

hyperbolehyperbolehyperbole

root of the functionroot of the functionroot of the function

set of values of a functionset of values of a functionset of values of a function

the greatest value of a function in a closed intervalthe greatest value of a function in a closed intervalthe greatest value of a function in a closed interval

the least value of a function in a closed intervalthe least value of a function in a closed intervalthe least value of a function in a closed interval