Topicmeed99b783a1f7204_1528449000663_0Topic

The symmetrysymmetrysymmetry of a graph of a function with respect to axis OY

Levelmeed99b783a1f7204_1528449084556_0Level

Third

Core curriculummeed99b783a1f7204_1528449076687_0Core curriculum

I. The functions. The student:

12) On the basis of function$y=f\left(x\right)$ plots the graphs of functions $y=f(x-a)$, $y=f\left(x\right)+b$, $y=-f\left(x\right)$,
$y=f(-x)$.

Timingmeed99b783a1f7204_1528449068082_0Timing

45 minutes

General objectivemeed99b783a1f7204_1528449523725_0General objective

Interpretation and the use of information presented both in a mathematical and popular science texts also using graphs, diagrams and tables.

Specific objectivesmeed99b783a1f7204_1528449552113_0Specific objectives

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

2. Recognizing and obtaining the graphs of functions in symmetrysymmetrysymmetry with relation to axis OY.

3. Getting to know the formula of a function, whose graph was obtained in symmetry with relation to axis OY.

Learning outcomesmeed99b783a1f7204_1528450430307_0Learning outcomes

The student:

- recognizes the graphs of functions obtained in symmetry with relation to axis OY,symmetry with relation to axis OYsymmetry with relation to axis OY,

- recognizes the formulae of functions, whose graphs were obtained in symmetrysymmetrysymmetry with relation to axis OY.

Methodsmeed99b783a1f7204_1528449534267_0Methods

1. Diamond ranking.

2. Situation analysis.

Forms of workmeed99b783a1f7204_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmeed99b783a1f7204_1528450127855_0Introduction

The students work in two groups using the diamond ranking technique. One group gets the information about the function in order, and the other group gets the information about the symmetrysymmetrysymmetry in the coordinate system in order.

When they have finished, the representatives of both groups present their posters.

The groups exchange information to verify and complete their knowledge.

Proceduremeed99b783a1f7204_1528446435040_0Procedure

The teacher informs the students that the aim of the class is getting to know the properties of the symmetrysymmetrysymmetry of the function with relation to axis OY.

Task
The students, working in groups, use their information about the symmetrysymmetrysymmetry of the function with relation to axis OY. They consider how the formula of a function will change if its graph will be symmetrically transformed with relation to axis OY. The students formulate hypotheses, analyse and check them watching the multimedia presentation. They formulate their conclusions.

[Slideshow]

Conclusion:

Transforming the graph of function f in symmetry with relation to axis OY, we obtain the graph of function h, described with formula $h\left(x\right)=f(-x)$.meed99b783a1f7204_1527752263647_0Transforming the graph of function f in symmetry with relation to axis OY, we obtain the graph of function h, described with formula $h\left(x\right)=f(-x)$.

The students use the information to solve the tasks individually.

Task
Function f is described with the following table.

[Table]

Make a table presenting the function described with formula $g\left(x\right)=f(-x)$.

Discussion – the domain of functiondomain of functiondomain of function $f\left(x\right)$ is set , the set of outputsset of outputsset of outputs , what is the domain and the set of outputs for function $g\left(x\right)=f(-x)$?

The conclusion that should be formulated by the students:

If the domain of function $f\left(x\right)$ is set , the set of outputs is and $g\left(x\right)=f(-x)$, then the domain of function g is set , and the set of outputs is set .meed99b783a1f7204_1527752256679_0If the domain of function $f\left(x\right)$ is set , the set of outputs is and $g\left(x\right)=f(-x)$, then the domain of function g is set , and the set of outputs is set .

The students do the task using the information.

Task
Function described with formula $f\left(x\right)=x^3-4$, where . Give the formula of function h, whose graph his symmetric about the graph of functiongraph of functiongraph of function f with relation to axis OY. Give the domain of function h. Plot graphs of both functions in one coordinate system.

Task
The domain of functiondomain of functiondomain of function f is set , and the set of outputsset of outputsset of outputs is set . Give the domain and the set of outputs of functionset of outputs of functionset of outputs of function g described with formula $g\left(x\right)=f(-x)$.

Task
Function ftakes as the smallest output (- 5) for x = 7 and takes the largest output 8 for x = - 2. Give the largest output, the smallest output and the inputs for which function $g\left(x\right)=f(-x)$ takes these outputs.

Having finished all the tasks, the students present their results. The teacher assesses their work and explains any doubts.

An extra task
The roots of function f are numbers 7 and (– 3). The graph of functiongraph of functiongraph of function g is symmetrical about the graph of function f with relation to axis OY. Calculate the value of expression $5g\left(3\right)+4g(-7)$.

Lesson summarymeed99b783a1f7204_1528450119332_0Lesson summary

The students do the consolidation tasks.

Then, they summarize the class and formulate the conclusions to be remembered.

- Transforming the graph of function f in symmetry with relation to axis OY, we obtain the graph of function h, described with formula $h\left(x\right)=f(-x)$.
- If the domain of function $f\left(x\right)$ is set , the set of outputs is and $g\left(x\right)=f(-x)$, then the domain of function g is set , and the set of outputs is set .meed99b783a1f7204_1527712094602_0- Transforming the graph of function f in symmetry with relation to axis OY, we obtain the graph of function h, described with formula $h\left(x\right)=f(-x)$.
- If the domain of function $f\left(x\right)$ is set , the set of outputs is and $g\left(x\right)=f(-x)$, then the domain of function g is set , and the set of outputs is set .

Selected words and expressions used in the lesson plan

domain of functiondomain of functiondomain of function

functionfunctionfunction

graph of functiongraph of functiongraph of function

pointpointpoint

set of outputsset of outputsset of outputs

set of outputs of functionset of outputs of functionset of outputs of function

symmetrysymmetrysymmetry

symmetry with relation to axis OYsymmetry with relation to axis OYsymmetry with relation to axis OY