Consider the most common transformations of graphs of trigonometric functions. Transformations of graphs of trigonometric functions with the module Graphs of trigonometric functions and their transformations
SUBJECT: Transformations of graphs of trigonometric functions with modulus.
TARGET: Consideration of obtaining graphs of trigonometric functions of the form
y= f(|x|) ;y = | f(x)| .
Develop mathematical logic and attention.
DURING THE CLASSES:
Org. moment: Announcement of the topic, goals and objectives of the lesson.
Teacher: Today we must learn how to graph functions y = sin |x|; y = cos|x|
Y = |A sin x +b| ; Y = |A cos x +b| using our knowledge of transformations of transcendental functions of the form y = f(|x|) and y = |f(x)| . You may ask, “What is this for?” The fact is that the properties of the functions change in this case, but this is best seen, as you know, on the graph.
Let's remember how these functions are written using the definition
Children: f(|x|) =
|f(x)| = 
Teacher: So, to plot the function y =f(|x|), if the graph of the function is known
y =f{ x), you need to leave that part of the graph of the function y = in placef(x), which
corresponds to the non-negative part of the domain of definition of the function y =f(x). Reflecting this
part is symmetrical about the y-axis, we get another part of the graph corresponding
negative part of the domain of definition.
That is, on the graph it looks like this: y = f (x)
(These graphs are drawn on the board. Children in notebooks)
Now, based on this, we will construct a graph of the functions y = sin |x|; Y = |sin x | ; Y = |2 sin x + 2|
Fig 1. Y = sin x
Figure 2. Y = sin |x|
Now let's plot the functions Y = |sin x | and Y = |2 sin x + 2|
To plot the function y = \f(x)\, if the graph of the function y = is knownf(x), you need to leave in place that part wheref(x) > ABOUT, and symmetrically display its other part relative to the x-axis, wheref(x) < 0.
Summary of algebra lesson and beginning of analysis in 10th grade
on the topic: “Transformation of graphs of trigonometric functions”
The purpose of the lesson: to systematize knowledge on the topic “Properties and graphs of trigonometric functions y=sin (x), y=cos (x)”.
Lesson objectives:
- repeat the properties of trigonometric functions y=sin (x), y=cos (x);
- repeat reduction formulas;
- converting graphs of trigonometric functions;
- develop attention, memory, logical thinking; intensify mental activity, the ability to analyze, generalize and reason;
- fostering hard work, diligence in achieving goals, interest in the subject.
Lesson equipment: ICT
Lesson type: learning new things
During the classes
Before the lesson, 2 students draw graphs from their homework on the board.
Organizing time:
Hello guys!
Today in the lesson we will transform the graphs of trigonometric functions y=sin (x), y=cos (x).
Oral work:
Checking homework.
solving puzzles.
Learning new material
All transformations of function graphs are universal - they are suitable for all functions, including trigonometric ones. Here we will limit ourselves to a brief reminder of the main transformations of graphs.
Transformation of function graphs.
The function y = f (x) is given. We start building all graphs from the graph of this function, then we perform actions with it.
Function
What to do with the schedule
y = f(x) + a
We raise all the points of the first graph by a units up.
y = f(x) – a
We lower all the points of the first graph down a units.
y = f(x + a)
We shift all points of the first graph by a units to the left.
y = f (x – a)
We shift all points of the first graph by a units to the right.
y = a*f (x),a>1
We fix the zeros in place, move the upper points higher by a times, and lower the lower ones lower by a times.
The graph will “stretch” up and down, the zeros remain in place.
y = a*f(x), a<1
We fix the zeros, the upper points will go down a times, the lower ones will rise a times. The graph will “shrink” towards the x-axis.
y = -f(x)
Mirror the first graph about the x-axis.
y = f (ax), a<1
Fix a point on the ordinate axis. Each segment on the abscissa axis is increased by a times. The graph will stretch from the ordinate axis in different directions.
y = f (ax), a >1
Fix a point on the ordinate axis, reduce each segment on the abscissa axis by a factor. The graph will “shrink” towards the y-axis on both sides.
y = | f(x)|
The parts of the graph located under the x-axis are mirrored. The entire graph will be located in the upper half-plane.
Solution schemes.
1)y = sin x + 2.
We build a graph y = sin x. We raise each point of the graph upward by 2 units (zeros too).
2)y = cos x – 3.
We build a graph y = cos x. We lower each point of the graph down by 3 units.
3)y = cos (x - /2)
We build a graph y = cos x. We shift all points by p/2 to the right.
4)y = 2 sinx.
We build a graph y = sin x. We leave the zeros in place, raise the upper points by 2 times, and lower the lower ones by the same amount.
PRACTICAL WORK Plotting graphs of trigonometric functions using the Advanced Grapher program.
Let's plot the function y = -cos 3x + 2.
- Let's plot the function y = cos x.
- Let's reflect it relative to the abscissa axis.
- This graph must be compressed three times along the x-axis.
- Finally, such a graph must be raised up by three units along the y-axis.
y = 0.5 sin x.
y = 0.2 cos x-2
y = 5cos 0 .5 x
y= -3sin(x+π).
2) Find the mistake and fix it.
V. Historical material. A message about Euler.
Leonhard Euler is the greatest mathematician of the 18th century. Born in Switzerland. For many years he lived and worked in Russia, a member of the St. Petersburg Academy.
Why should we know and remember the name of this scientist?
By the beginning of the 18th century, trigonometry was still not sufficiently developed: there were no symbols, formulas were written in words, it was difficult to learn them, the question of the signs of trigonometric functions in different quarters of a circle was unclear, and the argument of a trigonometric function meant only angles or arcs. Only in the works of Euler did trigonometry receive its modern form. It was he who began to consider the trigonometric function of a number, i.e. Argument began to be understood not only as arcs or degrees, but also as numbers. Euler derived all trigonometric formulas from several basic ones and streamlined the question of the signs of the trigonometric function in different quarters of the circle. To denote trigonometric functions, he introduced the symbolism: sin x, cos x, tan x, ctg x.
At the threshold of the 18th century, a new direction appeared in the development of trigonometry - analytical. If before this the main goal of trigonometry was considered to be the solution of triangles, then Euler considered trigonometry as the science of trigonometric functions. The first part: the doctrine of functions is part of the general doctrine of functions, which is studied in mathematical analysis. Part two: solving triangles - geometry chapter. Such innovations were made by Euler.
VI. Repetition
Independent work “Add the formula.”
VII. Lesson summary:
1) What new did you learn in class today?
2) What else do you want to know?
3) Grading.
Algorithm for constructing graphs The graph of the function y = sin (x-a) can be obtained by parallel moving the graph of the function y = sinx along the Ox axis by a units to the right. The graph of the function y = sin (x+a) can be obtained by parallel moving the graph of the function y = sinx along the Ox axis by a units to the left.
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0) can be obtained from the graph of the function y = sin x by stretching it (for K>1 by stretching by a factor of K) along the Oy axis. The graph of the function y = Кsin (x) (К>0) can be obtained from the graph of the function y = sinx its с" title="Graphing algorithm: The graph of the function y = Кsin (x) (К>0) can be obtained from the graph of the function y = sin x by stretching it (for K>1 by stretching by K times) along the Oy axis. The graph of the function y = Ksin (x) (K>0) can be obtained from the graph of the function y = sinx it with" class="link_thumb"> 9 !} Algorithm for constructing graphs: The graph of the function y = Ksin (x) (K>0) can be obtained from the graph of the function y = sin x by stretching it (for K>1 by stretching it by a factor of K) along the Oy axis. The graph of the function y = Кsin (x) (К>0) can be obtained from the graph of the function y = sinx by compressing it (at 01 stretching by K times) along the Оу axis. The graph of the function y = Ksin (x) (K>0) can be obtained from the graph of the function y = sinx its c "> 0) can be obtained from the graph of the function y = sin x by stretching it (for K>1 by stretching K times) along the axis Oy. The graph of the function y = Ksin (x) (K>0) can be obtained from the graph of the function y = sinx by compressing it (with 01 stretching by K times) along the Oy axis. Graph of the function y = Ksin (x) (K>0) can be obtained from the graph of the function y = sinx it with" title=" Algorithm for constructing graphs: The graph of the function y = Ksin (x) (K>0) can be obtained from the graph of the function y = sin x by stretching it (for K> 1 stretching by K times) along the Oy axis. The graph of the function y = Ksin (x) (K>0) can be obtained from the graph of the function y = sinx with it"> title="Algorithm for constructing graphs: The graph of the function y = Ksin (x) (K>0) can be obtained from the graph of the function y = sin x by stretching it (for K>1 by stretching it by a factor of K) along the Oy axis. The graph of the function y = Кsin (x) (К>0) can be obtained from the graph of the function y = sinx it with">!}
1 stretch 0 1 stretch 0 10 10 Compression and stretching to the x-axis K > 1 stretching 0 1 stretching 0 1 stretching 0 1 stretching 0 1 stretching 0 title="10 Compression and stretching to the x-axis K > 1 stretching 0
13 Shift along the ordinate axis Build a graph of the function y=sins+3 Build a graph of the function y=sins-3 + up - down y = sinx y = sinx + 3 y = sinx y = sinx Transformation of the graph
X y 1 -2 Check: y 1 = sinx; y 2 = sinx + 2; y 3 = sinx
Algebra lesson notes in 10th grade
Vasilyeva Ekaterina Sergeevna,
mathematic teacher
OGBOU "Smolensk special (correctional)
comprehensive school of types I and II"
Smolensk
Lesson topic: “Transformation of graphs of trigonometric functions.”
Namemodule: converting graphs of trigonometric functions. Integratingdidactictarget: practice skills in constructing graphs of trigonometric functions. Target action plan for students:
- review the basic properties of trigonometric functions; practice the skill of converting graphs of trigonometric functions; promote the development of logical thinking; cultivate interest in studying the subject.
Bank of information.
Incoming control. Name the properties of the functions y = sin x (Fig. 1).Rice. 1
Properties:
- D(y)=R E(y)=[-1;1], the function is limited sin(-x)=-sinx, the function is odd Minimum positive period: 2π
sin (x+2πn)= sin x, n Є Z, x Є R. sin x=0 at x=πk, kЄ Z sin x>0, x Є (2πk;2π+2πk), k Є Z sin x Greatest the value equal to 1, y=sin x takes at the points x=π/2+ 2πk, k Є Z. The smallest value equal to -1, y=sin x takes at the points x=3π/2+ 2πk, k Є Z.
Rice. 2
Properties:
- D (y)=R E (y)=[-1;1], the function is limited cos(-x)= cos x, the function is even Minimum positive period: 2π
cos (x+2πn)=cos x, n Є Z, x Є R cos x=0 at x=π/2+πk, kЄZ cos x>0, x Є (-π/2+2πk; π/2+ 2πk), k Є Z cos x The largest value equal to 1, y=cos x takes at points x= 2πk, k Є Z. The smallest value equal to -1, y=cos x takes at points x=π+ 2πk, k Є Z.
Rice . 3
Properties:
- D(y)-set of all real numbers, except numbers of the form x=π/2 +πk, k Є Z E(y)=(-∞;+ ∞), unbounded function tg(-x)=-tg x, odd function smallest positive period: π
tg(x+π)= tan x tgx= 0 at x=πk, k Є Z tg x> 0, x Є (πk; π/2+πk), k Є Z tg x
Rice. 4
Properties:
- D(y)-set of all real numbers, except numbers of the form x=πk, k Є Z E(y)= (-∞;+ ∞), unbounded function ctg(-x)=-ctg x, odd function Minimum positive period: π
ctg(x+π)=tg x ctg x = 0 at x=π/2+πk, k Є Z ctg x>0, x Є(πk; π/2+πk), k Є Z ctg x
Explanation of the material.
- y=
f(x)+
a, where a is a constant number, you need to move the graph y=
f(x)
along the ordinate axis. If a>0, then we move the graph parallel to itself upward, if a To construct a graph of the function y=
kf(x)
we need to stretch the graph of the function y=
f(x)
V k
times along the ordinate axis. If |
k|>1
, then the graph stretches along the axis OY, If 0k| , then – compression. Graph of a function y=
f(x+
b)
obtained from the graph y=
f(x)
by parallel translation along the abscissa axis. If b>0, then the graph moves to the left, if b
To graph a function y= f(kx) need to stretch the schedule y= f(x) along the abscissa axis. If | k|>1 , then the graph is compressed along the axis OH, if 0
Fixing the material.
Level A
Privatedidactictarget: practice the skill of constructing trigonometric functions through transformations.
Methodicala commentForstudents:
Ox 3 times.
The graph of a function is obtained from a graph by stretching along the axis Oy 2 times.
The graph of a function is obtained from the graph by parallel translation 2 units up along the axis Oy.
The graph of a function is obtained from the graph by parallel translation along the abscissa axis by units to the left.
G 
The graph of a function is obtained from the graph by compressing along the axis Oy 4 times.
Level B.
Privatedidactictarget: trigonometric functions by consistent applying transformations.
Methodicala commentForstudents: construct graphs of functions by performing transformations.
The graph of a function is obtained from the graph by parallel translation along the abscissa axis by units to the right.
The graph of a function is obtained from the graph of a function by sequentially performing the following transformations:
1) parallel translation by units to the left along the abscissa axis
2) compression along the Oy axis by 4 times .
The graph of the function is obtained from the graph of the function, each ordinate of which changes by a factor of -2. To do this, we perform the following transformations:
1) display symmetrically about the axis Ox,
2) stretch 2 times along the axis Oy.
consistent perform the following transformations:
1) compression along the abscissa axis by 2 times;
2) stretching V 3 times along axes Oy;
3) parallel transfer on 1 unit up along axes ordinate.
Level WITH .
Privatedidactictarget: practice graphing skills trigonometric functions by consistent applying transformations.
Methodical a comment For students : please indicate , which transformation need to execute For construction graphs . Build graphics .
1. 
The graph of a function is obtained from the graph of a function by sequentially performing the following transformations:
1) display is symmetrical about the axis Ox,
2) compression by 2 times along the Oy axis;
3) parallel translation 2 units down along the Oy axis.
2.
The graph of a function is obtained from the graph of a function consistent performing the following transformations: it turns out www. aiportal. ru/ services/ graph. html
