Plotting trigonometric functions in grade 11
Mathematics teacher of the first qualification category of the MAOU "Gymnasium No. 37", Kazan
Spiridonova L.V.
- Trigonometric functions of a numeric argument
- y = sin (x) + m and y = cos (x) + m
- Plotting functions of the form y = sin (x + t) and y = cos (x + t)
- Plotting functions of the form y = A · sin (x) and y = A · cos (x)
- Examples of
Trigonometric functions a numeric argument.
y = sin (x)
y = cos (x)
Plotting a function y = sin x .
Plotting a function y = sin x .
Plotting a function y = sin x .
Plotting a function y = sin x .
Properties of the function y = sin ( x ) .
all real numbers ( R )
2. Range of changes (Range of values) , E (y) = [ - 1; 1 ] .
3. Function y = sin ( x) odd, because sin (- x ) = - sin x
- π .
sin (x + 2 π ) = sin (x).
5. Continuous function
Decreases: [ π /2; 3 π /2 ] .
6. Increases: [ - π /2; π /2 ] .
+
+
+
-
-
-
Plotting a function y = cos x .
Function graph y = cos x obtained by transfer
the graph of the function y = sin x left on π /2.
Properties of the function y = co s ( x ) .
1. The domain of the function is the set
all real numbers ( R )
2. Range of changes (Range of values), E (y) = [ - 1; 1 ] .
3. Function y = cos (X) even, because cos (- X ) = cos (X)
- The function is periodic, with a main period of 2 π .
cos ( X + 2 π ) = cos (X) .
5. Continuous function
Decreases: [ 0 ; π ] .
6. Increases: [ π ; 2 π ] .
+
+
+
+
-
-
-
Building
charts functions of the form
y = sin ( x ) + m
and
y = cos (X) + m.
0, or down if m. "Width =" 640 " Parallel transfer of the graph along the Oy axis
Function graph y = f (x) + m is obtained by parallel transfer of the function graph y = f (x) , up on m units if m 0 ,
or down if m .
0 y m 1 x "width =" 640 " Conversion: y = sin ( x ) + m
Shift y = sin ( x ) along the axis y up if m 0
m
0 y m 1 x "width =" 640 " Conversion: y = cos ( x ) + m
Shift y = cos ( x ) along the axis y up , if m 0
m
Conversion: y = sin ( x ) + m
Shift y = sin ( x ) along the axis y down, if m 0
m
Conversion: y = cos ( x ) + m
Shift y = cos ( x ) along the axis y down if m 0
m
Building
charts functions of the form
y = sin ( x + t )
and
y = cos ( X + t )
0 and to the right if t is 0. "width =" 640 " Parallel transfer of the graph along the Ox axis
Function graph y = f (x + t) is obtained by parallel transfer of the function graph y = f (x) along the axis X on the | t | scale units to the left, if t 0
and to the right , if t 0.
0 y 1 x t "width =" 640 " Conversion: y = sin (x + t)
shift y = f (x) along the axis X to the left, if t 0
t
0 y 1 x t "width =" 640 " Conversion: y = cos (x + t)
shift y = f (x) along the axis X to the left, if t 0
t
Conversion: y = sin (x + t)
shift y = f (x) along the axis X to the right, if t 0
t
Conversion: y = cos (x + t)
shift y = f (x) along the axis X to the right, if t 0
t
0
1 and 0 a 1 "width =" 640 " Plotting functions of the form y = A · sin ( x ) and y = A · cos ( x ) , at a 1 and 0 a 1
1 and shrinking to the Ox axis with a factor of 0 A. "width =" 640 " Compression and stretching along the Ox axis
Function graph y = A · f (x ) we obtain by stretching the graph of the function y = f (x) with a factor A along the Ox axis if A 1 and compression to the Ox axis with a factor of 0 A .
1 let a = 1.5 y 1 x -1 "width =" 640 " Conversion: y = a sin ( x ), a 1
let a = 1.5
1 let a = 1.5 y 1 x "width =" 640 " Conversion: y = a · cos ( x ), a 1
let a = 1.5
Conversion: y = a sin ( x ) , 0
let a = 0.5
Conversion: y = a cos ( x ), 0
let a = 0.5
sin (
y
x
y = sin (x) → y = sin (x- π )
x
sin (
y
y
sin (
x
y
x
- 1
y = cos (x) → y = cos (2x) → y = - cos (2x) → y = - cos (2x) +3
x
x
x
y
y
sin
y
sin
sin
sin
y
x
y
x
- 1
y = sin (x) → y = sin (x / 3) → y = sin (x / 3) -2
y
x
- 1
y = sin (x) → y = 2sin (x) → y = 2sin (x) -1
y
y
y
cos
y
cos x + 2
x
cos x + 2
cos x
y
x
- 1
y = cos (x) → y = 1/2 cos (x) → y = -1 / 2 cos (x) → y = -1 / 2 cos (x) +2
y
x
- 1
y = cos (x) → y = cos (2x) → y = - cos (2x) →
Abstract of the algebra lesson and the beginning of the analysis in grade 10
on the topic: "Transforming 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 the casting formulas;
- transformation of graphs of trigonometric functions;
- develop attention, memory, logical thinking; to intensify mental activity, the ability to analyze, generalize and reason;
- education of hard work, diligence in achieving goals, interest in the subject.
Lesson equipment: ict
Lesson type: learning new
During the classes
Before the lesson, 2 students build graphs from their homework on the blackboard.
Organizing time:
Hello guys!
Today in the lesson we will transform the graphs of trigonometric functions y = sin (x), y = cos (x).
Oral work:
Homework check.
solving puzzles.
Learning new material
All transformations of graphs of functions are universal - they are suitable for all functions, including trigonometric ones. Here we restrict ourselves to a brief reminder of the main transformations of graphs.
Convert function graphs.
A function y = f (x) is given. We begin to build 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
All points of the first chart are raised by a units up.
y = f (x) - a
We lower all points of the first graph by a units down.
y = f (x + a)
All points of the first graph are shifted by a units to the left.
y = f (x - a)
All points of the first graph are shifted by a units to the right.
y = a * f (x), a> 1
We fix the zeros in place, move the top points higher by a times, and lower the lower ones lower by a times.
The graph "stretches" 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 go up a times. The graph will "shrink" to the abscissa axis.
y = -f (x)
Mirror the first graph about the abscissa axis.
y = f (ax), a<1
Anchor the point on the ordinate axis. Increase each segment on the abscissa by a factor. The graph will stretch from the ordinate axis in different directions.
y = f (ax), a> 1
Fix a point on the ordinate axis, decrease each segment on the abscissa axis by a times. The graph will "shrink" to the ordinate axis on both sides.
y = | f (x) |
The parts of the graph located under the abscissa axis should be 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 up by 2 units (zeroes too).
2)y = cos x - 3.
We build a graph y = cos x. We move each point of the graph down by 3 units.
3)y = cos (x - / 2)
We build a graph y = cos x. All points are shifted n / 2 to the right.
4) y = 2 sin x.
We build a graph y = sin x. We leave the zeros in place, raise the upper points 2 times, lower the lower ones by the same amount.
PRACTICAL WORK Plotting trigonometric functions using the Advanced Grapher program.
Let's build a graph of the function y = -cos 3x + 2.
- Let's build a graph of the function y = cos x.
- Let's reflect it about the abscissa axis.
- This graph must be compressed three times along the abscissa axis.
- Finally, such a graph should be raised up by three units along the ordinate axis.
y = 0.5 sin x.
y = 0.2 cos x-2
y = 5cos 0 , 5 x
y = -3sin (x + π).
2) Find the error and fix it.
V. Historical material. A post about Euler.
Leonard Euler is the greatest mathematician of the 18th century. Was 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 insufficiently developed: there were no conventional symbols, formulas were written in words, it was difficult to assimilate them, the question of the signs of trigonometric functions in different quarters of the circle was also unclear, and only angles or arcs were understood as the argument of a trigonometric function. It was only in the writings of Euler that trigonometry received its modern form. It was he who began to consider the trigonometric function of a number, i.e. the argument came to be understood not only as arcs or degrees, but also 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 symbols: sin x, cos x, tg x, ctg x.
On the threshold of the 18th century, a new direction appeared in the development of trigonometry - the analytical one. If before that 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 function is part of the general doctrine of functions, which is studied in mathematical analysis. Second part: solving triangles - chapter of geometry. Such innovations were made by Euler.
Vi. Repetition
Independent work "Add the formula".
Vii. Lesson summary:
1) What new have you learned in the lesson today?
2) What else do you want to know?
3) Grading.
ALGEBRA
Lessons for 10 grades
Topic.Plotting trigonometric functions
The purpose of the lesson: plotting the functions y = sin x, y = cos x, y = tg x, y = ctg x.
Formation of the ability to build graphs of functions: y = Asin (kx + b), y = Acos (kx + b), y = Atg (kx + b), y = Actg (kx + b).
I. Checking homework
1. One student reproduces the solution to exercise number 24 (1-3).
2. Frontal conversation:
1) Name the phenomena in nature that are periodically repeated.
2) Give the definition of a periodic function.
3) If the function y = f (x) has a period of the number T, then the period of this function will be the number 2T, 3T ...? Justify the answer.
4) Find the smallest positive period of the functions:
a) y = cos; b) y = sin; c) y = tg; d) y =.
5) periodic function y = C? If yes, please indicate the period of this function.
II. Plotting the function y = sin x
To plot the function y = sin x, we use the unit circle. Let's build a unit circle with a radius of 1 cm (2 cells). On the right, let's build a coordinate system, as in Fig. 57.
Draw points on the OX axis; π; ; 2 π (respectively 3 cells, 6 cells 9 cells, 12 cells). Let's divide the first quarter of the unit circle into three equal parts and into the same number of parts the segment of the abscissa axis. Move the sine value to the corresponding points on the OX axis. We get the points that need to be connected with a smooth line. Then we divide the second, third and fourth quarters of the unit circle also into three equal parts and transfer the sine value to the corresponding point on the OX axis. Sequentially connecting all the points obtained, we get a graph of the function y = sin x on the interval.
Since the function y = sin x is periodic with a period of 2 π, then to plot the function y = sin x on the entire line OX, it is enough to parallelly move the plot along the OX axis by 2 π, 4 π, 6 π ... units to the left and to the right (fig. 58).
A curve that is a graph of a function y = sin x is called a sinusoid.
Exercise ______________________________
1. Build graphs of functions.
a) y = sin; b) y = sin 2x; c) y = 2 sin x; d) y = sin (-x).
Answers: a) fig. 59; b) Fig. 60; c) Fig. 61; d) Fig. 62.
III. Plotting the function y = cos x
As you know, cos x = sin, so y = cos x and y = sin are the same functions. To plot the graph of the function y = sin, we will use the geometric transformations of the graphs: first, we will build (Fig. 63) the graph of the function y = sin x, then y = sin (-x) and at the end y = sin.
Exercise ________________________________
1. Plot the graphs of the functions:
a) y = cos; b) y = cos; c) y = cos x; d) y = | cos x |.
Answer: a) fig. 64; b) Fig. 65; c) Fig. 66; d) Fig. 67.
IV. Plotting the function y = tg x
The graph of the function y = tg x is constructed using the tangent line on the interval, the length of which is equal to the period π of this function. Construct a unit circle with a radius of 2 cm (4 cells) and draw a line of tangents. On the right, let's build a coordinate system, as in Fig. 68.
Draw points on the OX axis; (6 cells). Divide the first and fourth quarters of the circle into 3 equal parts and into the same number of parts each of the segments and. Find the values of the tangents of the numbers; ; 0; ; using a line of tangents (ordinates of points;;;; lines of tangents). Let's transfer the values of the tangents to the corresponding points of the OX axis. Sequentially connecting all the points obtained, we get a graph of the function y = tg x on the interval.
Since the function y = tg x is periodic with a period of π, to plot the function y = tg x on the entire line OX, it is enough to parallelly move the plot along the OX axis by π, 2 π, 3 π, 4 π ... units to the left and to the right (fig. 69).
The graph of the function y = tg x is called tangential.
Exercise
1. Plot the functions
a) y = tg 2x; b) y = t gx; c) y = tan x + 2; d) y = tg (-x).
Answers: a) fig. 70; b) Fig. 71; c) Fig. 72; d) Fig. 73.
V. Plotting the function y = ctg x
The graph of the function y = ctg x is easy to obtain using the formula ctg x = tg and two geometric transformations (Fig. 74) symmetry about the ΟΥ axis, parallel transfer along the OX axis by.
IV. Homework
Section I § 6. Questions and tasks for repeating section I No. 50-51. Exercises number 28 (a-d).
V. Lesson summary
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Slide captions:
Graphs of trigonometric functions Function y = sin x, its properties Conversion of graphs of trigonometric functions by parallel transfer Conversion of graphs of trigonometric functions by compression and expansion For the curious ...
trigonometric functions The graph of the function y = sin x is a sinusoid Properties of the function: D (y) = R Periodic (T = 2 ) Odd (sin (-x) = - sin x) Zeros of the function: y = 0, sin x = 0 at x = n, n Z y = sin x
trigonometric functions Properties of the function y = sin x 5. Intervals of constant sign: Y> 0 for x (0 + 2 n; +2 n), n Z Y
trigonometric functions Properties of the function y = sin x 6. Intervals of monotonicity: the function increases on intervals of the form: - / 2 +2 n; / 2 + 2 n n Z y = sin x
trigonometric functions Properties of the function y = sin x Intervals of monotonicity: the function decreases on intervals of the form: / 2 +2 n; 3 / 2 + 2 n n Z y = sin x
trigonometric functions Properties of the function y = sin x 7. Extremum points: X max = / 2 +2 n, n Z X m in = - / 2 +2 n, n Z y = sin x
trigonometric functions Properties of the function y = sin x 8. Range of values: E (y) = -1; 1 y = sin x
trigonometric functions Transformation of graphs of trigonometric functions The graph of the function y = f (x + b) is obtained from the graph of the function y = f (x) by parallel translation by (-b) units along the abscissa axis The graph of the function y = f (x) + a is obtained from the graph function y = f (x) by parallel translation by (a) units along the ordinate axis
trigonometric functions Convert graphs of trigonometric functions Build a graph Functions y = sin (x + / 4) remember the rules
trigonometric functions Convert the graphs of trigonometric functions y = sin (x + / 4) Plot the function: y = sin (x - / 6)
trigonometric functions Convert graphs of trigonometric functions y = sin x + Plot the function: y = sin (x - / 6)
trigonometric functions Convert graphs of trigonometric functions y = sin x + Plot the function: y = sin (x + / 2) remember the rules
trigonometric functions The graph of the function y = cos x is the cosine. List the properties of the function y = cos x sin (x + / 2) = cos x
trigonometric functions Transformation of graphs of trigonometric functions by compression and expansion The graph of the function y = kf (x) is obtained from the graph of the function y = f (x) by stretching it k times (for k> 1) along the ordinate Graph of the function y = kf (x ) is obtained from the graph of the function y = f (x) by compressing it by a factor of k (at 0
Trigonometric Functions Shrink and Stretch the graphs of trigonometric functions y = sin2x y = sin4x Y = sin0.5x remember the rules
trigonometric functions Transformation of graphs of trigonometric functions by compression and expansion The graph of the function y = f (kx) is obtained from the graph of the function y = f (x) by compressing it k times (for k> 1) along the abscissa axis Graph of the function y = f (kx ) is obtained from the graph of the function y = f (x) by stretching it k times (at 0
trigonometric functions Convert trigonometric function graphs by compressing and stretching y = cos2x y = cos 0.5x remember the rules
trigonometric functions Transformation of graphs of trigonometric functions by compression and expansion Graphs of functions y = -f (kx) and y = - kf (x) are obtained from graphs of functions y = f (kx) and y = kf (x), respectively, by mirroring them with respect to the abscissa axis sine is an odd function, therefore sin (-kx) = - sin (kx) cosine is an even function, therefore cos (-kx) = cos (kx)
trigonometric functions Convert graphs of trigonometric functions by compressing and stretching y = - sin3x y = sin3x remember the rules
trigonometric functions Convert graphs of trigonometric functions by compressing and stretching y = 2cosx y = -2cosx remember the rules
trigonometric functions Transformation of graphs of trigonometric functions by compression and expansion The graph of the function y = f (kx + b) is obtained from the graph of the function y = f (x) by transferring it in parallel by (-v / k) units along the abscissa axis and by compressing into k times (for k> 1) or stretching by a factor of k (for 0
Trigonometric Functions Shrink and stretch graphs of trigonometric functions Y = cos (2x + / 3) y = cos (x + / 6) y = cos (2x + / 3) y = cos (2 (x + / 6)) y = cos (2x + / 3) y = cos (2 (x + / 6)) Y = cos (2x + / 3) y = cos2x remember the rules
trigonometric functions For the curious ... Check out how the graphs of some other triggers look like. functions: y = 1 / cos x or y = sec x (read the sec) y = cosec x or y = 1 / sin x read the cosec
On the subject: methodological developments, presentations and notes
CRC "Transformation of graphs of trigonometric functions" 10-11 grades
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