"Yoshkar-Ola College of Service Technologies"
Building and researching a graph trigonometric function y=sinx in spreadsheetMS excel
/methodological development/
Yoshkar - Ola
Subject. Construction and study of the graph of a trigonometric functiony = sinx in spreadsheet MS Excel
Lesson type– integrated (acquisition of new knowledge)
Goals:
Didactic goal - explore the behavior of graphs of a trigonometric functiony= sinxdepending on the coefficients using a computer
Tutorials:
1. Find out the change in the graph of the trigonometric function y= sin x depending on coefficients
2. Show implementation computer technology in teaching mathematics, the integration of two subjects: algebra and computer science.
3. To form the skills of using computer technology in mathematics lessons
4. Strengthen the skills of researching functions and plotting their graphs
Developing:
1. Develop cognitive interest students to academic disciplines and the ability to apply their knowledge in practical situations
2. Develop the ability to analyze, compare, highlight the main thing
3. Contribute to the improvement of the overall level of development of students
educators :
1. Cultivate independence, accuracy, diligence
2. Foster a culture of dialogue
Forms of work in the lesson - combined
Didactic equipment and equipment:
1. Computers
2. Multimedia projector
4. Handout
5. Presentation slides
During the classes
I. Organization of the beginning of the lesson
Greeting students and guests
· Prepare for the lesson
II. Goal-setting and actualization of the topic
It takes a lot of time to study a function and build its graph, you have to perform a lot of cumbersome calculations, this is not convenient, computer technologies come to the rescue.
Today we will learn how to plot trigonometric functions in the environment spreadsheet processor MS Excel 2007.
The topic of our lesson is “Construction and study of the graph of a trigonometric function y= sinx in a spreadsheet"
From the course of algebra, we know the scheme for studying a function and constructing its graph. Let's remember how to do it.
slide 2
Function Study Scheme
1. Function domain (D(f))
2. Value area of the function Е(f)
3. Definition of parity
4. Periodicity
5. Function zeros (y=0)
6. Intervals of constant sign (y>0, y<0)
7. Intervals of monotonicity
8. Function extremes
III. Primary assimilation of new educational material
Open MS Excel 2007.
Let's plot the function y=sin x
Plotting in a spreadsheetMS excel 2007
The graph of this function will be built on the segment xЄ [-2π; 2π]
We will take the values of the argument with a step , to make the graph more accurate.
Since the editor works with numbers, let's convert radians to numbers, knowing that P ≈ 3.14 . (translation table in the handout).
1. Find the value of the function at the point x \u003d -2P. For the rest, the editor automatically calculates the corresponding function values for the corresponding values of the argument.
2. Now we have a table with argument and function values. With this data, we have to plot this function using the Chart Wizard.
3. To build a graph, you need to select the desired data range, rows with argument values and functions
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We write the conclusions in a notebook (Slide 5)
Conclusion. The graph of the function of the form y=sinx+k is obtained from the graph of the function y=sinx using parallel translation along the y-axis by k units
If k >0, then the graph is shifted up by k units
If k<0, то график смещается вниз на k единиц
Construction and study of the view functiony=k*sinx,k- const
Task 2. At work Sheet2 plot functions in one coordinate system y= sinx y=2* sinx, y= * sinx, on the interval (-2π; 2π) and see how the graph changes.
(In order not to re-set the value of the argument, let's copy the existing values. Now you need to set the formula, and build a graph using the resulting table.)
We compare the obtained graphs. We analyze together with the students the behavior of the graph of the trigonometric function depending on the coefficients. (Slide 6)
https://pandia.ru/text/78/510/images/image005_66.gif" width="16" height="41 src=">x , on the interval (-2π; 2π) and see how the graph changes.
We compare the obtained graphs. We analyze together with the students the behavior of the graph of the trigonometric function depending on the coefficients. (Slide 8)
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We write the conclusions in a notebook (Slide 11)
Conclusion. The graph of the function of the form y \u003d sin (x + k) is obtained from the graph of the function y \u003d sinx using parallel translation along the OX axis by k units
If k >1, then the graph is shifted to the right along the OX axis
If 0 IV. Primary consolidation of acquired knowledge Differentiated cards with a task to build and study a function using a graph Y=6*sin(x) Y=1-2
sinX Y=-
sin(3x+)
1.
Domain 2.
Scope of value 3.
Parity 4.
Periodicity 5.
Constancy intervals 6.
gapsmonotony Function rises Function decreasing 7.
Function extremes Minimum Maximum V. Homework organization Plot the function y=-2*sinх+1 , investigate and check the correctness of the construction in the Microsoft Excel spreadsheet environment. (Slide 12) VI. Reflection Now we will consider the question of how to plot the trigonometric functions of multiple angles ωx, where ω
is some positive number. To plot a function y = sin ωx Let's compare this function with the function we have already studied y = sinx. Let's assume that at x = x 0
function y = sin x takes a value equal to 0 . Then y 0 = sin x 0
. Let's transform this ratio as follows: Therefore, the function y = sin ωx at X = x
0
/ ω
takes the same value at 0
, which is the function y = sin x at x = x 0
. And this means that the function y = sin ωx repeats its values in ω
times more often than the function y = sin x. So the graph of the function y = sin ωx obtained by "compressing" the graph of the function y = sinx in ω
times along the x-axis. For example, the graph of the function y \u003d sin 2x obtained by "compressing" the sinusoid y = sinx twice along the abscissa. Function Graph y \u003d sin x / 2
obtained by "stretching" the sinusoid y \u003d sin x twice (or "compressing" in 1 /
2
times) along the x-axis. Since the function y = sin ωx repeats its values in ω
times more often than the function It is interesting to study the behavior of the function y \u003d sin ax on the example of animation, which can be very easily created in the program maple: Similarly, graphs are constructed for other trigonometric functions of multiple angles. The figure shows a graph of the function y = cos 2x, which is obtained by "compressing" the cosine y = cos x twice along the x-axis. Function Graph y = cos x / 2
obtained by "stretching" the cosine wave y = cos x twice along the x-axis. In the figure you see a graph of the function y = tg 2x, obtained by "compressing" the tangentoid y = tg x twice along the abscissa. Function Graph y = tg x / 2
, obtained by "stretching" the tangentoid y = tg x twice along the x-axis. And finally, the animation performed by the program maple: Exercises
1.
Build graphs of these functions and indicate the coordinates of the points of intersection of these graphs with the coordinate axes. Determine the periods of these functions. a). y=sin 4x / 3
G). y=tg 5x / 6
g). y = cos 2x / 3
b). y= cos 5x / 3
e). y=ctg 5x / 3
h). y=ctg x / 3
in). y=tg 4x / 3
e). y = sin 2x / 3
2.
Define Function Periods y \u003d sin (πx) and y = tg (πх / 2). 3.
Give two examples of a function that takes all values from -1 to +1 (including these two numbers) and changes periodically with a period of 10. 4
*. Give two examples of functions that take all values from 0 to 1 (including these two numbers) and change periodically with a period π / 2. 5.
Give two examples of functions that take all real values and change periodically with period 1. 6
*. Give two examples of functions that take all negative values and zero, but do not take positive values and change periodically with a period of 5. How to plot the function y=sin x? First, consider the graph of the sine on the interval. We take a single segment with a length of 2 cells of a notebook. We mark the unit on the Oy axis. For convenience, we round the number π/2 to 1.5 (and not to 1.6, as required by the rounding rules). In this case, a segment of length π/2 corresponds to 3 cells. On the Ox axis, we mark not single segments, but segments of length π / 2 (every 3 cells). Accordingly, a segment of length π corresponds to 6 cells, a segment of length π/6 corresponds to 1 cell. With this choice of a single segment, the graph shown on a sheet of notebook in a box corresponds to the graph of the function y=sin x as much as possible. Let's make a table of sine values on the interval: The resulting points are marked on the coordinate plane: Since y=sin x is an odd function, the sine graph is symmetrical with respect to the origin - point O(0;0). Taking into account this fact, we continue to plot the graph to the left, then the points -π: The function y=sin x is periodic with period T=2π. Therefore, the graph of the function, taken on the interval [-π; π], is repeated an infinite number of times to the right and left. Additional materials Manuals and simulators in the online store "Integral" for grade 10 from 1C
What will we study:
Guys, we have already met with the trigonometric functions of a numerical argument. Do you remember them? Let's take a closer look at the Y=sin(X) function Let's write down some properties of this function: 4) The function Y=sin(X) is bounded from below and above. This property comes from the fact that Let's use properties 1-5 to plot the function Y=sin(X). We will build our graph sequentially, applying our properties. Let's start building a graph on the segment. Particular attention should be paid to the scale. On the ordinate axis, it is more convenient to take a single segment equal to 2 cells, and on the abscissa axis - a single segment (two cells) to be taken equal to π / 3 (see figure). Let's calculate the values of the function on our segment: Let's build a graph for our points, taking into account the third property. Let's use the second property, which says that our function is odd, which means that it can be reflected symmetrically about the origin: We know that sin(x+ 2π) = sin(x). This means that on the interval [- π; π] graph looks the same as on the segment [π; 3π] or or [-3π; - pi] and so on. It remains for us to carefully redraw the graph in the previous figure on the entire x-axis. The graph of the function Y=sin(X) is called a sinusoid. Let's write a few more properties according to the constructed graph: 1. Solve the equation sin(x)= x-π Solution: Let's build 2 graphs of the function: y=sin(x) and y=x-π (see figure). 2. Plot the function y=sin(π/6+x)-1 Solution: The desired graph is obtained by moving the graph of the function y=sin(x) by π/6 units to the left and 1 unit down. Solution: Let's build a graph of the function and consider our segment [π/2; 5π/4].
y = sinx, then its period in ω
times less than the period of the function y = sinx. For example, the period of the function y \u003d sin 2x equals 2π / 2 = π
, and the period of the function y \u003d sin x / 2
equals π
/
x / 2
= 4π .
Lesson and presentation on the topic: "Function y=sin(x). Definitions and properties"
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We solve problems in geometry. Interactive construction tasks for grades 7-10
Software environment "1C: Mathematical constructor 6.1"sine properties. Y=sin(X)
1) The domain of definition is the set of real numbers.
2) The function is odd. Let's recall the definition of an odd function. A function is called odd if the equality is true: y(-x)=-y(x). As we remember from ghost formulas: sin(-x)=-sin(x). The definition is satisfied, so Y=sin(X) is an odd function.
3) The function Y=sin(X) increases on the interval and decreases on the interval [π/2; π]. When we move along the first quarter (counterclockwise), the ordinate increases, and when we move along the second quarter, it decreases.
-1 ≤ sin(X) ≤ 1
5) The smallest value of the function is -1 (for x = - π/2+ πk). The largest value of the function is 1 (for x = π/2+ πk)._2.png)
Plotting the function sine x, y=sin(x)
_3.png)
_4.png)
Conversion table for ghost formulas
_5.png)
_6.png)
6) The function Y=sin(X) increases on any segment of the form: [- π/2+ 2πk; π/2+ 2πk], k is an integer and decreases on any segment of the form: [π/2+ 2πk; 3π/2+ 2πk], k is an integer.
7) The function Y=sin(X) is a continuous function. Let's look at the graph of the function and make sure that our function has no breaks, this means continuity.
8) Range of values: segment [- 1; one]. This is also clearly visible from the graph of the function.
9) The function Y=sin(X) is a periodic function. Let's look at the graph again and see that the function takes on the same values at some intervals.Examples of problems with sine
Our graphs intersect at one point A(π; 0), this is the answer: x = π_7.png)
_8.png)
The graph of the function shows that the largest and smallest values are reached at the ends of the segment, at the points π/2 and 5π/4, respectively.
Answer: sin(π/2) = 1 is the largest value, sin(5π/4) = the smallest value._9.png)
Sine problems for independent solution
