Graph of the function y \u003d sin x. Plot function y \u003d sin2x and y \u003d sin Benefits of plotting online

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Benefits of charting online

• Visual display of input functions
• Building very complex graphs
• Plotting implicitly (e.g. ellipse x ^ 2/9 + y ^ 2/16 \u003d 1)
• The ability to save charts and receive a link to them, which becomes available to everyone on the Internet
• Scale control, line color
• Possibility of plotting graphs by points, using constants
• Simultaneous construction of several graphs of functions
• Plotting in polar coordinates (use r and θ (\\ theta))

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How to plot the function y \u003d sin x? First, let's look at the sine graph in the interval.

We take a single segment with a length of 2 cells of a notebook. Mark one 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 unit 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 - 1 cell.

With this choice of a unit segment, the graph depicted on a sheet of a notebook in a box corresponds as much as possible to the graph of the function y \u003d sin x.

Let's make a table of sine values \u200b\u200bin the interval:

We mark the obtained points on the coordinate plane:

Since y \u003d sin x is an odd function, the sine graph is symmetrical about the origin - point O (0; 0). Taking this fact into account, we will continue to plot the graph to the left, then the points -π:

The function y \u003d sin x is periodic with a period T \u003d 2π. Therefore, the graph of the function, taken on the interval [-π; π], is repeated an infinite number of times to the right and to the left.

Lesson and presentation on the topic: "Function y \u003d sin (x). Definitions and properties"

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Manuals and simulators in the Integral online store for grade 10 from 1C
We solve problems in geometry. Interactive building tasks for grades 7-10
Software environment "1C: Mathematical Designer 6.1"

What we will study:

• Properties of the function Y \u003d sin (X).
• Function graph.
• How to build a graph and its scale.
• Examples.

Sine properties. Y \u003d sin (X)

Guys, we already got acquainted with trigonometric functions of a numeric argument. Do you remember them?

Let's take a closer look at the function Y \u003d sin (X)

Let's write some properties of this function:
1) Domain of definition - a set of real numbers.
2) The function is odd. Let's remember the definition of an odd function. The function is called odd if the equality holds: y (-x) \u003d - y (x). As we remember from the ghost formulas: sin (-x) \u003d - sin (x). The definition is fulfilled, so Y \u003d sin (X) is an odd function.
3) The function Y \u003d sin (X) increases on the segment and decreases on the segment [π / 2; π]. When we move along the first quarter (counterclockwise), the ordinate increases, and when we move along the second quarter, it decreases.

4) The function Y \u003d sin (X) is bounded from below and from above. This property follows from the fact that
-1 ≤ sin (X) ≤ 1
5) The smallest value of the function is -1 (at x \u003d - π / 2 + πk). The largest value of the function is 1 (at x \u003d π / 2 + πk).

Let's use properties 1-5 to graph the function Y \u003d sin (X). We will build our graph sequentially, applying our properties. Let's start building a graph on a segment.

Particular attention should be paid to the scale. On the ordinate axis it is more convenient to take a unit segment equal to 2 cells, and on the abscissa axis - to take a unit segment (two cells) equal to π / 3 (see figure).

Plot sine x function, y \u003d sin (x)

Let's calculate the values \u200b\u200bof the function on our segment:

Let's build a graph based on our points, taking into account the third property.

Conversion table for ghost formulas

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 sin (x + 2π) \u003d sin (x). This means that on the segment [- π; π] the graph looks the same as on the segment [π; 3π] or or [-3π; - π] and so on. It remains for us to carefully redraw the graph in the previous figure on the entire abscissa axis.

The graph of the function Y \u003d sin (X) is called a sinusoid.

Let's write a few more properties according to the constructed graph:
6) The function Y \u003d 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) Function Y \u003d sin (X) is a continuous function. Let's look at the graph of the function and make sure that our function has no discontinuities, which means continuity.
8) Range of values: segment [- 1; one]. This is also clearly seen from the function graph.
9) Function Y \u003d sin (X) is a periodic function. Let's look at the graph again and see that the function takes on the same values \u200b\u200bat some intervals.

Examples of sine problems

1. Solve the equation sin (x) \u003d x-π

Solution: Let's build 2 graphs of the function: y \u003d sin (x) and y \u003d x-π (see figure).
Our graphs intersect at one point A (π; 0), this is the answer: x \u003d π

2. Plot the function y \u003d sin (π / 6 + x) -1

Solution: The desired graph is obtained by moving the graph of the function y \u003d sin (x) by π / 6 units to the left and 1 unit down.

Solution: Let's plot the function and consider our segment [π / 2; 5π / 4].
The graph of the function shows that the largest and smallest values \u200b\u200bare reached at the ends of the segment, at points π / 2 and 5π / 4, respectively.
Answer: sin (π / 2) \u003d 1 is the largest value, sin (5π / 4) \u003d the smallest value.

Sine problems for independent solution

• Solve the equation: sin (x) \u003d x + 3π, sin (x) \u003d x-5π
• Graph the function y \u003d sin (π / 3 + x) -2
• Plot function y \u003d sin (-2π / 3 + x) +1
• Find the largest and smallest value of the function y \u003d sin (x) on an interval
• Find the largest and smallest value of the function y \u003d sin (x) on the segment [- π / 3; 5π / 6]