COS X p 2 schedule. Graphs of trigonometric functions of multiple corners

Lesson and presentation on the topic: "Function Y \u003d COS (X). Definition and graph of the function"

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Educational aids and simulators in the online store "Integral" for grade 10
Algebraic tasks with parameters, 9-11 classes
Software Wednesday "1C: Mathematical Designer 6.1"

What we will study:
1. Definition.
2. Function schedule.
3. The properties of the function y \u003d cos (x).
4. Examples.

Definition of the cosine function y \u003d cos (x)

Guys, we already got acquainted with the function y \u003d sin (x).

Let's remember one of the ghost formulas: Sin (X + π / 2) \u003d COS (X).

Thanks to this formula, we can argue that the functions sin (x + π / 2) and COS (X) are identical, and their graphics of functions coincide.

The graph of the SIN function (X + π / 2) is obtained from the graph of the SIN (X) function parallel transfer to π / 2 units left. This will be the graph of the function y \u003d cos (x).

The graph of the function y \u003d cos (x) is also called the sinusoid.

COS (X) function properties

The definition area is a lot of valid numbers.
Function is even. Let's remember the definition of an even function. The function is even if the equality y (-x) \u003d y (x) is performed. As we remember from the ghost formulas: COS (-X) \u003d - COS (X), the definition was executed, then the cosine is an even function.
The function y \u003d cos (x) decreases on the segment and increases on the segment [π; 2π]. In this we can make sure the chart of our function.
The function y \u003d cos (x) is limited to below and from above. This property follows from the fact that
-1 ≤ cos (x) ≤ 1
The smallest value of the function is -1 (at x \u003d π + 2πk). The greatest value Functions equal to 1 (at x \u003d 2πk).
The function y \u003d cos (x) is a continuous function. Let's look at the schedule and make sure that our function has no breaks, it means continuity.
The range of segments [- 1; one]. It is also clearly seen from the schedule.
The function y \u003d cos (x) is a periodic function. Let's see again on the schedule and see that the function takes the same values \u200b\u200bthrough some intervals.

Examples with COS (X) function

1. Solve the COS equation (x) \u003d (x - 2π) 2 + 1

Solution: Build 2 graphics of the function: y \u003d cos (x) and y \u003d (x - 2π) 2 + 1 (see Figure).

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

2. Build a graph of the function y \u003d cos (x) at x ≤ 0 and y \u003d sin (x) at x ≥ 0

Solution: To build the desired schedule, let's build two charts on "pieces". First piece: y \u003d cos (x) at x ≤ 0. second piece: y \u003d sin (x)
at x ≥ 0. We will show both "pieces" on one schedule.

3. Find the greatest and the smallest value functions y \u003d cos (x) on the segment [π; 7π / 4]

Solution: Build a function schedule and consider our segment [π; 7π / 4]. The graph shows that the greatest and smallest values \u200b\u200bare achieved at the ends of the segment: at points π and 7π / 4, respectively.
Answer: COS (π) \u003d -1 - the smallest value, COS (7π / 4) \u003d the greatest value.

4. Build a graph of the function y \u003d cos (π / 3 - x) + 1

Solution: COS (-X) \u003d COS (x), then the desired schedule will turn out by transferring the function of the function y \u003d cos (x) to π / 3 units to the right and 1 unit up.

Tasks for self solutions

1) Solve equation: cos (x) \u003d x - π / 2.
2) to solve the equation: cos (x) \u003d - (x - π) 2 - 1.
3) Construct a graph of the function y \u003d cos (π / 4 + x) - 2.
4) Build a graph of the function y \u003d cos (-2π / 3 + x) + 1.
5) Find the greatest and smallest value of the function y \u003d cos (x) on the segment.
6) find the largest and smallest value of the function y \u003d cos (x) on the segment [- π / 6; 5π / 4].

"Function graphics and their properties" - y \u003d CTG x. 4) limited function. 3) odd function. (The graph of the function is symmetrical on the start of the coordinates). Y \u003d TG x. 7) The function is continuous on any interval of the species (? K;? +? K). The function y \u003d TG x is continuous on any interval of the species. 4) The function decreases at any interval of the species (? K;? +? K). Function graph Y \u003d TG X is called a tangent.

"Function graph Y X" - parabola template y \u003d x2. To see graphics, click. Example 2. We construct a graph of the function y \u003d x2 + 1, based on the function of the function y \u003d x2 (click on the mouse). Example 3. We prove that the graph of the function y \u003d x2 + 6x + 8 is parabola, and we will build a schedule. The graph of the function y \u003d (x - m) 2 is a parabola with a vertex at the point (m; 0).

"Mathematics graphs" - how can I build graphs? The most naturally functional dependencies are reflected using graphs. Interesting application: drawings, ... Why are we studying graphics? Graphics elementary functions. What can you draw with graphs? We consider the application of graphs in training subjects: mathematics, physics, ...

"Building graphs with a derivative" - \u200b\u200bgeneralization. Build a sketch of the graphics of the function. Find asymptotes graph graphics. Schedule derivative function. Additional task. Explore the function. Call the launches of the decrease of the function. Independent work Pupils. Expand knowledge. Lesson fixing the studied material. Rate your skills. Points of maximum function.

"Graphs with a module" - display the "bottom" part into the upper half-plane. The module of the actual number. Properties function y \u003d | x |. | x |. Numbers. Algorithm for building a graphic function. Algorithm of construction. Function Y \u003d LXL. Properties. Independent work. Zero function. Tips of the Great. Decision of independent work.

"The equation of tangent" is the equation of tangent. The equation is normal. If, then the curves intersect at right angles. Terms of parallelism and perpendicularity of two straight lines. The angle between the graphs of functions. Equation tangent to function graphics at point. Let the function differentiate at the point. Let straight are given by equations and.

Total in the subject of 25 presentations

Now we will consider the question of how to build graphs trigonometric functions multiple corners ωx where ω - Some positive number.

To build a graph function y \u003d sin. ωx Compare this feature with the function already studied by us. y \u003d sin x. Suppose that when x \u003d X. 0 function y \u003d sin x Takes a value equal to 0. Then

0 \u003d sin x. 0 .

We transform this ratio as follows:

Consequently, the function y \u003d sin. ωx for h. = x. 0 / ω takes the same value w. 0 that function y \u003d sin x for x \u003d x. 0 . And this means that the function y \u003d sin. ωx repeats your meanings in ω times more often than function y \u003d sin x. Therefore, the function is a function y \u003d sin. ωx It turns out by "compression" graphics function y \u003d sin x in ω Once along the x axis.

For example, a function schedule y \u003d sin 2x It turns out by "compression" of sinusoids y \u003d sin x halved along the abscissa axis.

Schedule function y \u003d sin x / 2 It turns out by "stretching" sinusoids y \u003d sin x twice (or "compression" in 1 / 2 times) along the x axis.

Since function y \u003d sin. ωx repeats your meanings in ω times more often than function
y \u003d sin xthen the period of it in ω once less than the function period y \u003d sin x. For example, the function of the function y \u003d sin 2x Raven 2π / 2. = π and the period of the function y \u003d sin x / 2 Raven π / X / 2 = 4π. .

Interesting to study the behavior of the function y \u003d sin axon the example of the animation, which is very easy to create in the program Maple.:

Similarly, graphs and other trigonometric functions of multiple corners are built. The figure shows a graph of a function. y \u003d cos 2xwhich turns out by "compression" of cosine y \u003d cos x twice along the abscissa axis.

Schedule function y \u003d cos X / 2 It turns out by "stretching" cosine y \u003d cos x twice along the x axis.

In the figure you see a graph of the function y \u003d tg 2xobtained by "compression" tangentsoid y \u003d tg xhalved along the abscissa axis.

Schedule function y \u003d TG. X / 2 obtained by "stretching" tangentsoids y \u003d 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 intersection points of these graphs with the coordinate axes. Determine the periods of these functions.

but). y \u003d sin. 4x / 3 d). Y \u003d TG. 5x / 6 g). y \u003d cos. 2x / 3

b). y \u003d cos. 5x / 3 e). y \u003d Ctg. 5x / 3 h). y \u003d Ctg. X / 3

in). Y \u003d TG. 4x / 3 e). y \u003d sin. 2x / 3

2. Determine periods of functions y \u003d sin (π) and y \u003d TG. ( πh / 2.).

3. Give two examples of functions that take all values \u200b\u200bfrom -1 to +1 (including these two numbers) and change periodically with a period of 10.

4 * Give two examples of functions that take all values \u200b\u200bfrom 0 to 1 (including these two numbers) and change periodically with the period π / 2..

5. Bring two examples of functions that take all valid values \u200b\u200band change periodically with a period of 1.

6 * Bring two examples of functions that take all negative values \u200b\u200band zero, but do not take positive values \u200b\u200band change periodically with a period of 5.