Building charts online. How to Graph a Function Graph a Linear Function
Previously, we studied other functions, for example, a linear one, let us recall its standard form:
hence the obvious fundamental difference - in the linear function X stands in the first degree, and in that new function, which we are starting to study, X stands in the second degree.
Recall that the graph of a linear function is a straight line, and the graph of a function, as we will see, is a curve called a parabola.
Let's start by finding out where the formula came from. The explanation is this: if we are given a square with a side a, then we can calculate its area as follows:
If we change the length of the side of the square, then its area will change.
So, one of the reasons why the function is studied is given
Recall that the variable X is an independent variable, or argument, in the physical interpretation it can be, for example, time. Distance is, on the contrary, a dependent variable, it depends on time. A dependent variable or function is a variable at.
This is the law of correspondence, according to which each value X mapped to a single value at.
Any correspondence law must satisfy the requirement of uniqueness from argument to function. In a physical interpretation, this looks quite clear on the example of the dependence of distance on time: at each moment of time we are at a certain distance from the starting point, and it is impossible at the same time at time t to be both 10 and 20 kilometers from the start of the journey.
At the same time, each function value can be reached with several argument values.
So, we need to build a graph of the function, to do this, make a table. Then, according to the graph, examine the function and its properties. But even before plotting the graph, we can say something about its properties by the form of the function: it is obvious that at cannot take negative values, since
So let's make a table:
Rice. one
It is easy to note the following properties from the graph:
Axis at is the axis of symmetry of the graph;
The top of the parabola is the point (0; 0);
We see that the function only accepts non-negative values;
In the interval where 
the function is decreasing, but on the interval where the function is increasing;
The function acquires the smallest value at the vertex, 
;
There is no maximum value of the function;
Example 1
Condition:
Decision:
Insofar as X conditionally changes on a specific interval, we can say about the function that it increases and changes on the interval . The function has a minimum value and a maximum value on this interval
Rice. 2. Graph of the function y = x 2 , x ∈
Example 2
Condition: Find the largest and smallest value features:
Decision:
X changes over the interval, which means at decreases on the interval while and increases on the interval while .
So the limits of change X, and the limits of change at, which means that on this interval there is both a minimum value of the function and a maximum
Rice. 3. Graph of the function y = x 2 , x ∈ [-3; 2]
Let us illustrate the fact that the same value of a function can be achieved with several values of the argument.
We choose a rectangular coordinate system on the plane and plot the values of the argument on the abscissa axis X, and on the y-axis - the values of the function y = f(x).
Function Graph y = f(x) the set of all points is called, for which the abscissas belong to the domain of the function, and the ordinates are equal to the corresponding values of the function.
In other words, the graph of the function y \u003d f (x) is the set of all points in the plane, the coordinates X, at which satisfy the relation y = f(x).
On fig. 45 and 46 are graphs of functions y = 2x + 1 and y \u003d x 2 - 2x.
Strictly speaking, one should distinguish between the graph of a function (exact mathematical definition which was given above) and the drawn curve, which always gives only a more or less accurate sketch of the graph (and even then, as a rule, not the entire graph, but only its part located in the final part of the plane). In what follows, however, we will usually refer to "chart" rather than "chart sketch".
Using a graph, you can find the value of a function at a point. Namely, if the point x = a belongs to the scope of the function y = f(x), then to find the number f(a)(i.e. the function values at the point x = a) should do so. Need through a dot with an abscissa x = a draw a straight line parallel to the y-axis; this line will intersect the graph of the function y = f(x) at one point; the ordinate of this point will be, by virtue of the definition of the graph, equal to f(a)(Fig. 47).
For example, for the function f(x) = x 2 - 2x using the graph (Fig. 46) we find f(-1) = 3, f(0) = 0, f(1) = -l, f(2) = 0, etc.
A function graph visually illustrates the behavior and properties of a function. For example, from a consideration of Fig. 46 it is clear that the function y \u003d x 2 - 2x takes positive values when X< 0 and at x > 2, negative - at 0< x < 2; наименьшее значение функция y \u003d x 2 - 2x accepts at x = 1.
To plot a function f(x) you need to find all points of the plane, coordinates X,at which satisfy the equation y = f(x). In most cases, this is impossible, since there are infinitely many such points. Therefore, the graph of the function is depicted approximately - with greater or lesser accuracy. The simplest is the multi-point plotting method. It consists in the fact that the argument X give a finite number of values - say, x 1 , x 2 , x 3 ,..., x k and make a table that includes the selected values of the function.
The table looks like this:
Having compiled such a table, we can outline several points on the graph of the function y = f(x). Then, connecting these points with a smooth line, we get an approximate view of the graph of the function y = f(x).
However, it should be noted that the multi-point plotting method is very unreliable. In fact, the behavior of the graph between the marked points and its behavior outside the segment between the extreme points taken remains unknown.
Example 1. To plot a function y = f(x) someone compiled a table of argument and function values:
The corresponding five points are shown in Fig. 48.
Based on the location of these points, he concluded that the graph of the function is a straight line (shown in Fig. 48 by a dotted line). Can this conclusion be considered reliable? Unless there are additional considerations to support this conclusion, it can hardly be considered reliable. reliable.
To substantiate our assertion, consider the function
.
Calculations show that the values of this function at points -2, -1, 0, 1, 2 are just described by the above table. However, the graph of this function is not at all a straight line (it is shown in Fig. 49). Another example is the function y = x + l + sinx; its meanings are also described in the table above.
These examples show that in its "pure" form, the multi-point plotting method is unreliable. Therefore, to plot a given function, as a rule, proceed as follows. First, the properties of this function are studied, with the help of which it is possible to construct a sketch of the graph. Then, by calculating the values of the function at several points (the choice of which depends on the set properties of the function), the corresponding points of the graph are found. And, finally, a curve is drawn through the constructed points using the properties of this function.
We will consider some (the most simple and frequently used) properties of functions used to find a sketch of a graph later, but now we will analyze some commonly used methods for plotting graphs.
Graph of the function y = |f(x)|.
It is often necessary to plot a function y = |f(x)|, where f(x) - given function. Recall how this is done. By definition of the absolute value of a number, one can write
This means that the graph of the function y=|f(x)| can be obtained from the graph, functions y = f(x) as follows: all points of the graph of the function y = f(x), whose ordinates are non-negative, should be left unchanged; further, instead of the points of the graph of the function y = f(x), having negative coordinates, one should construct the corresponding points of the graph of the function y = -f(x)(i.e. part of the function graph
y = f(x), which lies below the axis X, should be reflected symmetrically about the axis X).
Example 2 Plot a function y = |x|.
We take the graph of the function y = x(Fig. 50, a) and part of this graph when X< 0 (lying under the axis X) is symmetrically reflected about the axis X. As a result, we get the graph of the function y = |x|(Fig. 50, b).
Example 3. Plot a function y = |x 2 - 2x|.
First we plot the function y = x 2 - 2x. The graph of this function is a parabola, the branches of which are directed upwards, the top of the parabola has coordinates (1; -1), its graph intersects the abscissa axis at points 0 and 2. On the interval (0; 2) the function takes negative values, therefore this part of the graph reflect symmetrically about the x-axis. Figure 51 shows a graph of the function y \u003d |x 2 -2x |, based on the graph of the function y = x 2 - 2x
Graph of the function y = f(x) + g(x)
Consider the problem of plotting the function y = f(x) + g(x). if graphs of functions are given y = f(x) and y = g(x).
Note that the domain of the function y = |f(x) + g(х)| is the set of all those values of x for which both functions y = f(x) and y = g(x) are defined, i.e. this domain of definition is the intersection of the domains of definition, the functions f(x) and g(x).
Let the points (x 0, y 1) and (x 0, y 2) respectively belong to the function graphs y = f(x) and y = g(x), i.e. y 1 \u003d f (x 0), y 2 \u003d g (x 0). Then the point (x0;. y1 + y2) belongs to the graph of the function y = f(x) + g(x)(for f(x 0) + g(x 0) = y 1+y2),. and any point of the graph of the function y = f(x) + g(x) can be obtained in this way. Therefore, the graph of the function y = f(x) + g(x) can be obtained from function graphs y = f(x). and y = g(x) by replacing each point ( x n, y 1) function graphics y = f(x) dot (x n, y 1 + y 2), where y 2 = g(x n), i.e., by shifting each point ( x n, y 1) function graph y = f(x) along the axis at by the amount y 1 \u003d g (x n). In this case, only such points are considered. X n for which both functions are defined y = f(x) and y = g(x).
This method of plotting a function graph y = f(x) + g(x) is called the addition of graphs of functions y = f(x) and y = g(x)
Example 4. In the figure, by the method of adding graphs, a graph of the function is constructed
y = x + sinx.
When plotting a function y = x + sinx we assumed that f(x) = x, a g(x) = sinx. To build a function graph, we select points with abscissas -1.5π, -, -0.5, 0, 0.5,, 1.5, 2. Values f(x) = x, g(x) = sinx, y = x + sinx we will calculate at the selected points and place the results in the table.
Build a function
We bring to your attention a service for plotting function graphs online, all rights to which belong to the company Desmos. Use the left column to enter functions. You can enter manually or using the virtual keyboard at the bottom of the window. To enlarge the chart window, you can hide both the left column and the virtual keyboard.
Benefits of online charting
- Visual display of introduced functions
- Building very complex graphs
- Plotting implicitly defined graphs (e.g. ellipse x^2/9+y^2/16=1)
- The ability to save charts and get a link to them, which becomes available to everyone on the Internet
- Scale control, line color
- The ability to plot graphs by points, the use of constants
- Construction of several graphs of functions at the same time
- Plotting in polar coordinates (use r and θ(\theta))
With us it is easy to build graphs of varying complexity online. The construction is done instantly. The service is in demand for finding intersection points of functions, for displaying graphs for their further transfer to a Word document as illustrations for solving problems, for analyzing the behavioral features of function graphs. The optimal browser for working with charts on this page of the site is Google Chrome. When using other browsers, correct operation is not guaranteed.
