Topicm886300914d6aebfe_1528449000663_0Topic

Shifting the function graph f(x)=a/x along the axis of the coordinate system.

Levelm886300914d6aebfe_1528449084556_0Level

Third

Core curriculumm886300914d6aebfe_1528449076687_0Core curriculum

V. Functions. Student:

12) sketches the graphs of the function y = f(x‑a), y = f(x) + b, y=-f(x), y=f(-x) on the basis of the graph function y = f(x).

Timingm886300914d6aebfe_1528449068082_0Timing

45 minutes

General objectivem886300914d6aebfe_1528449523725_0General objective

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

Specific objectivesm886300914d6aebfe_1528449552113_0Specific objectives

1. Communicating in English, developing mathematics skills, scientific, technical and IT competences; developing learning skills.

2. Drawing the graph of the function y = f(x‑a), y = f(x) + b.

3. Determining the properties of a function based on its graph.

Learning outcomesm886300914d6aebfe_1528450430307_0Learning outcomes

The student:

- draws the graph of the function y = f(x‑a), y = f(x) + b,

- determines the properties of a function based on its graph.

Methodsm886300914d6aebfe_1528449534267_0Methods

1. Competition.

2. Case study.

Forms of workm886300914d6aebfe_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm886300914d6aebfe_1528450127855_0Introduction

Competition.

Decide if the sentence is True or False.

a) The graph of the functiongraph of the functiongraph of the function y=2(x‑2)Indeks górny 2² is obtained by shifting the graph of the functiongraph of the functiongraph of the function y=2xIndeks górny 2² by 2 values to the right along the axis X.

b) The graph of the functiongraph of the functiongraph of the function y=-3xIndeks górny 2²-2 is obtained as a result of shifting the graph of the functiongraph of the functiongraph of the function y=-3xIndeks górny 2² o 2 by 2 values upwards along the axis X.

c) After shifting the function graph y=-2xIndeks górny 2² by three values to the left along the X axis, we obtain a graph of the functiongraph of the functiongraph of the function y=-2(x+3)Indeks górny 2².

d) After shifting the graph of the functiongraph of the functiongraph of the function y=xIndeks górny 2² by four values down along the X axis, we get a graph of the functiongraph of the functiongraph of the function y=xIndeks górny 2²-4.

The teacher summarizes the results of the competition and clarifies any doubts.

Procedurem886300914d6aebfe_1528446435040_0Procedure

The teacher informs students that the aim of the lesson will be to draw graphs of functions obtained as a result of shifting the graphshifting the graphshifting the graph of the functionfunctionfunction $y=\frac{a}{x}$ along the axis of the coordinate system.

Students working in groups prepare graphs of the functionfunctionfunction $y=\frac{a}{x}+q$. They formulate the appropriate conclusions.

Task

Draw a graph of the functiongraph of the functiongraph of the function

Group 1: $y=\frac{2}{x}+3$

Group 2: $y=-\frac{2}{x}+3$

What do you notice? Formulate the appropriate conclusion.

Conclusion

To make a graph of the functiongraph of the functiongraph of the function $y=\frac{a}{x}+q$, first we sketch the graph of the functiongraph of the functiongraph of the function $y=\frac{a}{x}$, and then we shift the obtained graph along the Y axis by $q$ values.

The asymptotes of the graphasymptotes of the graphasymptotes of the graph are straight lines: $y=q$ - a horizontal asymptoteasymptoteasymptote and $x=0$ - a vertical asymptoteasymptoteasymptote.

Students, working in groups, prepare graphs of the functions of the form $y=\frac{a}{x-p}$. They formulate the appropriate conclusions.

Task

Draw a graph of the following functionm886300914d6aebfe_1527752263647_0Draw a graph of the following function

Group 1: $y=\frac{2}{x-3}$

Group 2: $y=\frac{2}{x+1}$

What do you notice? Formulate the appropriate conclusion.m886300914d6aebfe_1527752263647_0What do you notice? Formulate the appropriate conclusion.

Conclusion

In order to make a graph of the functiongraph of the functiongraph of the function $y=\frac{a}{x-p}$, one should first draw the graph of the functiongraph of the functiongraph of the function

$y=\frac{a}{x}$, and then shift it along the axis X by $p$ values.

The asymptotes of the graphasymptotes of the graphasymptotes of the graph are straight lines: $x=p$ - a vertical asymptoteasymptoteasymptote, $y=0$ - a horizontal asymptoteasymptoteasymptote.

Students think how a graph of the functionfunctionfunction $y=\frac{a}{x-p}+q$ can be obtained. The material contained in the applet will help them. They formulate the appropriate conclusions.

[Geogebra applet]

Analyze the material contained in the applet. Describe the properties of the functionfunctionfunction $y=\frac{2}{x-2}+3$ on the basis of its graph. Formulate the appropriate conclusion.

Conclusion

To draw a functionfunctionfunction $y=\frac{a}{x-p}+q$, first draw the functionfunctionfunction $y=\frac{a}{x}$, then shift each point of this graph along the X axis by $p$ values and along the Y by $q$ values. The asymptotes of the graphasymptotes of the graphasymptotes of the graph are straight lines:

$x=p$ - a vertical asymptoteasymptoteasymptote, $y=q$ - a horizontal asymptoteasymptoteasymptote.

Using the information learned, students solve the tasks independently.

Task

Find the equations of the asymptotes of the functionfunctionfunction $f\left(x\right)=\frac{6}{x-4}-2$. Give its domain and set of values.

Task for volunteers

Find $q$, if point $P(\frac{2}{3},-4)$ belong to the functionfunctionfunction $f\left(x\right)=-\frac{6}{x}+q$.

Students do the revision exercises and formulate conclusions to remember.

Remember

- To draw a functionfunctionfunction $y=\frac{a}{x-p}+q$, first sketch the functionfunctionfunction $y=\frac{a}{x}$, then shift each point of this graph along the X axis by $p$ values and along the Y by $q$ values.

- The asymptotes of the graphasymptotes of the graphasymptotes of the graph are straight lines: $x=p$ - a vertical asymptoteasymptoteasymptote, $y=q$ - a horizontal asymptoteasymptoteasymptote.

Selected words and expressions used in the lesson plan

asymptoteasymptoteasymptote

asymptotes of the graphasymptotes of the graphasymptotes of the graph

functionfunctionfunction

graph of the functiongraph of the functiongraph of the function

shifting the graphshifting the graphshifting the graph