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

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Symetria wykresu funkcji względem osi OY

Learning objectives

You will develop the following competences: communicating in English, mathematical, IT and basic scientific and technical competence, your learning skills.

You will get to know how to recognize and obtain the graphs of functions in symmetry with relation to axis OY.

You will get to know the formula of a function, whose graph was obtained in symmetry with relation to axis OY.

Learning effect

You will learn to recognize the graphs of functions obtained in symmetry with relation to axis OY.
You will learn recognize the formulae of a functions, whose graphs were obtained in symmetry with relation to axis OY.

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nagranie abstraktu

Recollect your knowledge about the functionfunctionfunction and the symmetry in the coordinate system. The aim of the class is getting to know the properties of the symmetrysymmetrysymmetry of the function with relation to axis OY.

Task 1

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nagranie abstraktu

Using the information about the symmetrysymmetrysymmetry of the pointpointpoint with relation to axis OY, consider how the formula of a functionfunctionfunction will change if its graph will be symmetrically transformed with relation to axis OY. Check your hypotheses analyzing the material shown in the multimedia presentation. Formulate appropriate conclusions.

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nagranie abstraktu

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

Use the information to solve the tasks.

Task 2

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nagranie abstraktu

FunctionfunctionFunction f is described with the following table.


x	- 12	- 10	- 6	- 3	2	5	8
f (x)	25	20	10	8	- 5	- 10	- 12

Make a table presenting the functionfunctionfunction described with formula $g (x) = f (- x)$ .

Think – if the domain of functiondomain of functiondomain of function $f (x)$ is set $<a, b>$ , the set of outputsset of outputsset of outputs $<c, d>$ , what is the domain and the set of outputs for function $g (x) = f (- x)$ ?

Formulate your conclusion.

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nagranie abstraktu

If the domain of functiondomain of functiondomain of function $f (x)$ is set $<a, b>$ , the set of outputs is $<c, d>$ and $g (x) = f (- x)$ , then the domain of functiondomain of functiondomain of function g is set $<- b, - a>$ , and the set of outputsset of outputsset of outputs is set $<c, d>$ .

Do the task using the information.

Task 3

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nagranie abstraktu

Function f described with formula $f (x) = x^{3} - 4$ , where $x \in <- 1; 2>$ . Give the formula of functionfunctionfunction h, whose graph his symmetric about the graph of functiongraph of functiongraph of function f with relation to axis OY. Give the domain of functiondomain of functiondomain of function h. Plot graphs of both functions in one coordinate system.

Task 4

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nagranie abstraktu

The domain of functiondomain of functiondomain of function f is set $D_{f} = <- 5; 10>$ , and the set of outputsset of outputsset of outputs is set $Z W_{f} = <1; \infty>$ . Give the domain and the set of outputs of functionset of outputs of functionset of outputs of function g described with formula $g (x) = f (- x)$ .

Task 5

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nagranie abstraktu

FunctionfunctionFunction f takes 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 $g (x) = f (- x)$ takes these outputs.

Task 6

An extra task:

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nagranie abstraktu

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

Do the consolidation tasks.

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nagranie abstraktu

Transforming the graph of functiongraph of functiongraph of function f in symmetry with relation to axis OYsymmetry with relation to axis OYsymmetry with relation to axis OY, we obtain the graph of functiongraph of functiongraph of function h, described with formula $h (x) = f (- x)$ .
If the domain of functiondomain of functiondomain of function $y = f (x)$ is set $<a, b>$ , the set of outputs isset of outputsset of outputs is $<c, d>$ then the domain of functiondomain of functiondomain of function $g (x) = f (- x)$ , and the set of its outputs will be respectively set $<- b, - a>$ and $<c, d>$ .

Exercises

Exercise 1

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Determine which sentence is true.

The graphs of functions $f (x) = 3 x - 4$ and $g (x) = - 3 x - 6$ are symmetrical about axis OY.
The domain of function $y = f (x)$ is set $-4;8>$ . The domain of function $g (x) = f (- x)$ is set $-8;4>$ .
The graph of function $f (x) = \sqrt{5 x}$ is symmetrical with relation to axis OY about the graph of function $g (x) = - \sqrt{5 x}$ .

zadanie

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Exercise 2

Sketch the graph of function $f (x) = \{\begin{cases} 2 x + 3 for x \in (- \infty; - 3 > \\ x^{2} for x \in (- 3; \infty) \end{cases}$ .
Next, in the same coordinate system sketch the graph of function $g (x) = f (- x)$ . Give the formula and the domain of function g.

$g (x) = \{\begin{cases} - 2 x + 3 for x \in < 3; \infty) \\ x^{2} for x \in (- \infty; 3) \end{cases}$

Exercise 3

Describe in English how you will transform the graph of function $y = f (x)$ in symmetry with relation to axis OY.

Exercise 4

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Indicate which pairs of expressions or words are translated correctly.

funkcja - function
wykres funkcji - graph of the function
dziedzina funkcji - The domain of the function
symetria względem osi - set of outputs of the function
zbiór wartości funkcji - symmetry with relation to axis

Rzadanie

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Match Polish terms with their English equivalents.

funkcja
zbiór wartości funkcji
graph of the function
dziedzina funkcji
function
symmetry with relation to axis
wykres funkcji
domain of the function
set of outputs of the function
symetria względem osi

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Glossary

domain of function

dziedzina funkcji

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wymowa w języku angielskim: domain of function

function

funkcja

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wymowa w języku angielskim: function

graph of function

wykres funkcji

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wymowa w języku angielskim: graph of function

point

punkt

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wymowa w języku angielskim: point

set of outputs

zbiór wartości

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wymowa w języku angielskim: set of outputs

set of outputs of function

zbiór wartości funkcji

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wymowa w języku angielskim: set of outputs of function

symmetry

symetria

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wymowa w języku angielskim: symmetry

symmetry with relation to axis OY

symetria względem osi OY

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wymowa w języku angielskim: symmetry with relation to axis OY

Keywords

domain of functiondomain of functiondomain of function

functionfunctionfunction

graph of functiongraph of functiongraph of function

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

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

Scenariusz

Exercises

Glossary

Keywords