home biology tutor Graph of the function y 2 cos x. Graphs of trigonometric functions of multiple angles. Tasks for independent solution

Graph of the function y 2 cos x. Graphs of trigonometric functions of multiple angles. Tasks for independent solution

"Graphs of functions and their properties" - y = ctg x. 4) Limited function. 3) odd function. (The graph of the function is symmetrical about the origin). y = tgx. 7) The function is continuous on any interval of the form (?k; ? + ?k). The function y = tg x is continuous on any interval of the form. 4) The function decreases on any interval of the form (?k; ? + ?k). The graph of the function y \u003d tg x is called the tangentoid.

"Graph of function Y X" - Parabola template y \u003d x2. Click to see graphs. Example 2. Let's build a graph of the function y = x2 + 1, based on the graph of the function y=x2 (mouse click). Example 3. Let's prove that the graph of the function y \u003d x2 + 6x + 8 is a parabola, and build a graph. The graph of the function y=(x - m)2 is a parabola with a vertex at the point (m; 0).

"Mathematics of graphics" - How can you build graphs? The most natural functional dependencies are reflected with the help of graphs. An interesting application: drawings, ... Why do we study graphs? Graphs elementary functions. What can you draw with graphs? We consider the use of graphs in academic subjects: mathematics, physics, ...

"Graphing with the Derivative" - Generalization. Construct a sketch of the graph of the function. Find the asymptotes of the graph of the function. Graph of the derivative of a function. Additional task. Explore the function. Name the intervals of decreasing function. Independent work students. Expand knowledge. Lesson to consolidate the studied material. Rate your skills. Maximum points of the function.

"Charts with the module" - Display the "lower" part in the upper half-plane. Module real number. Properties of the function y = |x|. |x|. Numbers. Algorithm for constructing a graph of a function. Construction algorithm. Function y=lхl. Properties. Independent work. Function nulls. Great advice. Do-it-yourself solution.

"Tangential equation" - Tangent equation. Normal equation. If, then the curves intersect at right angles. Conditions of parallelism and perpendicularity of two lines. Angle between function graphs. The equation of the tangent to the graph of a function at a point. Let the function be differentiable at a point. Let the lines be given by the equations and.

There are 25 presentations in total in the topic

Now we will consider the question of how to build graphs trigonometric functions 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 = sin x. 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 = sin x in ω times along the x-axis.

For example, the graph of the function y \u003d sin 2x obtained by "compressing" the sinusoid y = sin x 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
y = sin x, then its period in ω times less than the period of the function y = sin x. 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 .

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 accept all actual values and change periodically with a period of 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.

Lesson and presentation on the topic: "Function y=cos(x). Definition and graph of a function"

Additional materials
Dear users, do not forget to leave your comments, feedback, suggestions. All materials are checked by an antivirus program.

Teaching aids and simulators in the online store "Integral" for grade 10
Algebraic problems with parameters, grades 9–11
Software environment "1C: Mathematical constructor 6.1"

What will we study:
1. Definition.
2. Graph of the function.
3. Properties of the function Y=cos(X).
4. Examples.

Definition of the cosine function y=cos(x)

Guys, we already got acquainted with the function Y=sin(X).

Let's remember one of ghost formulas: sin(X + π/2) = cos(X).

Thanks to this formula, we can assert that the functions sin(X + π/2) and cos(X) are identical, and their function graphs are the same.

The graph of the function sin(X + π/2) is obtained from the graph of the function sin(X) parallel transferπ/2 units to the left. This will be the graph of the function Y=cos(X).

The graph of the function Y=cos(X) is also called a sinusoid.

cos(x) function properties

The domain of definition is the set of real numbers.
The function is even. Let's remember the definition even function. A function is called even if the equality y(-x)=y(x) holds. As we remember from the ghost formulas: cos(-x)=-cos(x), the definition is fulfilled, then the cosine is an even function.
The function Y=cos(X) decreases on the interval and increases on the interval [π; 2π]. We can verify this on the graph of our function.
The function Y=cos(X) is bounded from below and above. This property comes from the fact that
-1 ≤ cos(X) ≤ 1
The smallest value of the function is -1 (for x = π + 2πk). Highest value function is equal to 1 (for x = 2πk).
The function Y=cos(X) is continuous function. Let's look at the graph and make sure that our function has no gaps, which means continuity.
The range of values is the segment [- 1; one]. This is also clearly visible from the graph.
The function Y=cos(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 with the cos(x) function

1. Solve the equation cos(X)=(x - 2π) 2 + 1

Solution: Let's build 2 graphs of the function: y=cos(x) and y=(x - 2π) 2 + 1 (see figure).

y \u003d (x - 2π) 2 + 1 is a parabola shifted to the right by 2π and up by 1. Our graphs intersect at one point A (2π; 1), this is the answer: x \u003d 2π.

2. Plot the function Y=cos(X) for x ≤ 0 and Y=sin(X) for x ≥ 0

Solution: To build the required graph, let's plot two graphs of the function piece by piece. First slice: y=cos(x) for x ≤ 0. Second slice: y=sin(x)
for x ≥ 0. Let's depict both "pieces" on one graph.

3. Find the largest and smallest value function Y=cos(X) on the segment [π; 7π/4]

Solution: Let's build a graph of the function and consider our segment [π; 7π/4]. The graph shows that the largest and smallest values are achieved at the ends of the segment: at the points π and 7π/4, respectively.
Answer: cos(π) = -1 is the smallest value, cos(7π/4) = the largest value.

4. Plot the function y=cos(π/3 - x) + 1

Solution: cos(-x)= cos(x), then the desired graph will be obtained by moving the graph of the function y=cos(x) π/3 units to the right and 1 unit up.

Tasks for independent solution

1) Solve the equation: cos (x) \u003d x - π / 2.
2) Solve the equation: cos(x)= - (x - π) 2 - 1.
3) Plot the function y=cos(π/4 + x) - 2.
4) Plot the function y=cos(-2π/3 + x) + 1.
5) Find the largest and smallest value of the function y=cos(x) on the segment .
6) Find the largest and smallest value of the function y=cos(x) on the interval [- π/6; 5π/4].