Tip 1: How to build a plot of shifts and deformations
Tip 1: How to build a plot of shifts and deformations
To construct a graph of a complex function,it is necessary to first compile a table of numerical values of a variable. It is much easier to construct it purely geometrically, by means of shifts and deformations.
Instructions
1
To plot the graph by shifts anddeformations, look at the function carefully and select the main part, the graph of which will be relatively easy to draw (according to the table of values). For example, in the function y = 3sin (x-n / 2), the main part is y = sinx, and it's simpler to start plotting y = 2√ (x-3) from the graph y = √ x.
2
Create a table of numerical values for a variable for a simplified function and plot the graph in the coordinate system. Then start to bring it to its original form.
3
To obtain a graph of a function of the type y = f (x-a)(for example, y = cos (x + n) or y = (x - 1) ^ 3, move it along the abscissa axis (as a rule, ox) by a distance a, while the line will move to the left for a0 and to the right for a ˃0.
4
If the number is added to the function, not to the argumenty = f (x) + b (for example, y = tgx + 5 or y = 2 + √ x), move the graph along the y-axis, that is, oy. For b˃0, slide the graph upward to the required number of units, and for b˂0, downward.
5
To construct a graph of the form y = Af (x) (for example,y = 5cosx or y = 6√x), the main graph must be stretched or compressed along the axis oy. In this case, each value of the function will increase by A times. The graph will shrink if А˂1 and stretched, if А˃1. If, in addition, A˂0, then additionally reflect the graph along the vertical symmetrically about the axis ox.
6
In the event that the variable x is multiplied by a numberdirectly under the sign of the function, that is, it has the form y = f (kx) (for example, y = √5x or y = sin3x), act in the same way. That is, extend the graph with respect to the x axis at k˂1, compress at k˃1. If k˂0, then reflect it horizontally with respect to the axis oy (since all the values of the argument will change the sign to the opposite one).
7
For a complex function combining severalthe listed changes, build the schedule consecutively. Start with the transformations that deform the graph (tapering or stretching), at the end, carry out the transfer to the required distance. Intermediate graphics do not erase, but draw another color, or a dashed line, sign each of them.
Tip 2: How to draw a function graph
The course of algebra and mathematical analysis assumes a fundamental study of functions, finding its limits, values at different points, differentiation and integration, and constructing graphs. The graph allows you to visualize the change function depending on the change in the argument.
Instructions
1
Since any function is a linear or nonlinear dependence on the argument, try to represent the function in the standard form y = f (x), where f (x) is a function, x is an argument, and y is the value function. Thus, to each concrete value of the argument there corresponds a specific value function.
2
Find the domain of definition function, as well as intersection points function with the abscissa and ordinate axes. To do this, calculate the value function at x = 0, then calculate for which value of the argument value function will be zero.
3
Explore the function for symmetry. The function will be even if for each x from its domain the equality f (-x) = f (x) holds, and odd if the inequality f (-x) = -f (x) is satisfied. It is also necessary to determine the frequency function. If for each x from the domain of definition function the equality f (T + x) = f (x) holds, where T is the period function, then it is considered periodic. These functions include function f (x) = sin (x), f (x) = cos (x), and so on.
4
Identify break points function, If there are any. Construct vertical, horizontal and inclined asymptotes.
5
Find the derivative function, and then the extremum points (maximum and minimum function). Equate the derivative to zero and find the abscissa of the extremum point. Then substitute it in equation function and find the ordinate of the extremum point. Find the intervals in which the function is monotone (decreases or increases over the entire interval).
6
Explore the function of the second derivative in order to determine the points of inflection function. To do this, equate the second derivative function to zero and find the abscissa of the inflection point function. The ordinate can be found by substituting this value into equation function.
7
Draw on paper in a cage or on a millimeterpaper are mutually perpendicular to the x and y coordinate axes, which intersect at a point with coordinates (0; 0). Postpone all found in the research process function points in the coordinate system. To schedule function was depicted more precisely, calculate the values function, substituting a few more values of the argument. Connect the points obtained by a smooth line (straight line or curve). For accurate construction of the schedule, use the templates.
Tip 3: How to plot cos
Schedule functions y = cos (x) can be constructed from points corresponding to standard values. This procedure will facilitate the knowledge of certain properties of the indicated trigonometric function.
You will need
- - millimeter paper,
- - a pencil,
- - ruler,
- - Trigonometric tables.
Instructions
1
Draw the coordinate axes X and Y. Sign them, set the dimension in the form of divisions at regular intervals. Put on the axes single values and specify the origin point O.
2
Mark the points that correspond to the valuescos 0 = cos 2? = cos -2? = 1, then, through the half-period of the function, designate the points cos? / 2 = cos 3/2 = cos -? / 2 = cos -3? / 2 = 0; then, after a half-period of the function, mark the points cos? = cos -? = -1, and also indicate the values of the function cos? / 6 = cos -? / 6 = / 2 on the graph, mark the standard tabulated values cos? / 4 = cos -? / 4 = / 2, and finally, find the points that correspond to the values cos? / 3 = cos -? / 3 =?.
3
When building a chart, consider the followingconditions. The function y = cos (x) vanishes at x =? (n + 1/2), where n? Z. It is continuous on the whole domain of definition. On the interval (0,? / 2), the function y = cos (x) decreases from 1 to 0, and the values of the function are positive. On the interval (? / 2,?) Y = cos (x) decreases from 0 to -1, while the values of the function are negative. On the interval (?, 3? / 2) y = cos (x) increases from -1 to 0, while the values of the function are negative. On the interval (3? / 2, 2?) Y = cos (x) increases from 0 to 1, while the values of the function are positive.
4
Denote the maximum of the function y = cos (x) at the points xmax = 2? N and the minimum at the points xmin =? + 2? N.
5
Connect all the points together with a smooth line. The result is a cosine wave - a graphical representation of this function.
