How to build a schedule?

In order to understand how to build a graphfunction, it is necessary to investigate the range of the value (the admissible values ​​of the function y (x)) and the domain of definition (admissible values ​​of the argument x). The simplest restrictions are the presence of roots, a trigonometric function, or fractions with a variable in the denominator in the expression.

Now let's see what the function is clear or fuzzy, check the function with respect to the coordinate axes. Another function can be periodic, when the components of the graph are repeated.

You also need to investigate the function at the intersection withaxes of coordinates, if such intersections exist, they should be noted on the graph. After this, we find the asymptotes of the graph of the function - inclined and vertical.

The vertical asymptotes can be found by means ofthe investigation of the points of discontinuity on the right and the left, and it is necessary to look for inclined asymptotes separately for minus infinity and separately for plus infinity the ratios of the function to x, in other words to find the limit of f (x) / x. If this limit is finite, then this is the coefficient k from the equation of the tangent y = kx + b. To find b, it is important to find the limits of infinity from the difference f (x) -kx. Now, substitute the value of b in the tangent equation. In the case where b or k can not be found, the limit either does not exist, or it is equal to infinity, and there are no asymptotes either.

Now, we need to find the first derivative of the function. For this, we need to find the value of the function at the points of the extremum, determining the regions of monotonic decrease and increase of the function.

If the function is greater than zero at each point of the interval, then on this interval the function increases. If the function is less than zero at each point of the interval, then on this interval the function decreases.

When the derivative passes through the point x0 with a changesign from plus to minus, then this point will become the maximum point. When the derivative passes through the point x0 with a change of sign from minus to plus, this point becomes the minimum point.

Now we need to find the second derivative, orin other words the first derivative of the first derivative. It will help to reveal concavity or convexity, as well as points of inflection. We find the values ​​of the function at these inflection points.

If the function is greater than zero at each point of the interval, then on this interval the function will be concave. If the function is less than zero at each point of the interval, then on this interval the function is convex.

How to build a line chart

A linear graph is a broken line thatallows you to see and compare indicators. It is important not to confuse the linear graph with the graph of the linear function, because their purpose and construction are very different.

To construct a linear graph,draw a coordinate plane, specify the names of the axes and units of measurement. On the abscissa, we mark the middle of the intervals, usually in the form of intervals there are intervals of time-year, quarter, month, day, hour, and so on.

On the y-axis, we find the values ​​that will becorrespond to the first interval, and at the intersection we put a point. In the same way, we mark the remaining points of the linear graph. Then we connect all the points obtained and obtain a linear graph in the form of a broken line.

How to construct a graph of a quadratic function

The graph of the quadratic function looks like this: y = A · x?+ B · x + C. Before you begin to build such a graph, you need to analytically investigate the function. Most often, the parabola plot, as it is also called, is built in a rectangular coordinate system, with two perpendicular axes Ox and Oy.

First, we write down the domain of the function definition. A parabola is defined on the entire number line if there are no conditions in the job. Most often, the domain is the set of real numbers.

Now we find the vertex of the parabola. Substitute the coordinate value along the abscissa axis in the equation and calculate the coordinates of the vertex along the ordinate axis. The found point is marked on the drawing.

Compare the coefficients with zero to understand the direction of the branches of the parabola. If the coefficient is greater than zero, the parabola will be directed upwards, if the coefficient is less than zero, down.

We find the set of values ​​of the function. When the parabola branches go up, all values ​​will be above zero. When the branches are directed down, the values ​​of the function will be below zero.

Now look for the zeros of the function,intersecting the coordinate axes. To do this, you need to equate x to zero and also calculate y. You also need to find out at what value of the argument the function y will be zero. And, note the points on the graph.

Find additional points for plotting. We make all the values ​​in the form of a table. In the first line we write down the values ​​of the argument x, and the second value of the function y.

Now, you know how to build a chart and you will not be difficult to draw any kind of chart.

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