Online charting. How to graph a function Plotting points on a coordinate plane

Build function

We offer to your attention a service for constructing graphs of functions 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 window with the graph, you can hide both the left column and the virtual keyboard.

Benefits of online charting

• Visual display of entered functions
• Building very complex graphs
• Construction of graphs specified implicitly (for example, ellipse x^2/9+y^2/16=1)
• The ability to save charts and receive a link to them, which becomes available to everyone on the Internet
• Control of scale, line color
• Possibility of plotting graphs by points, using constants
• Plotting several function graphs simultaneously
• Plotting in polar coordinates (use r and θ(\theta))

With us it’s easy to build charts of varying complexity online. Construction is done instantly. The service is in demand for finding intersection points of functions, for depicting graphs for further moving them into a Word document as illustrations when solving problems, and for analyzing the behavioral features of function graphs. The optimal browser for working with charts on this website page is Google Chrome. Correct operation is not guaranteed when using other browsers.

Previously, we studied other functions, for example linear, let us recall its standard form:

hence the obvious fundamental difference - in the linear function X stands in the first degree, and in the new function we are beginning to study, X stands to the second power.

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 side A, then we can calculate its area like this:

If we change the length of the side of a square, then its area will change.

So, this is one of the reasons why the function is studied

Recall that the variable X- this is an independent variable, or argument; in a physical interpretation, it can be, for example, time. Distance is, on the contrary, a dependent variable; it depends on time. The dependent variable or function is a variable at.

This is the law of correspondence, according to which each value X a single value is assigned at.

Any correspondence law must satisfy the requirement of uniqueness from argument to function. In a physical interpretation, this looks quite clear using 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 to be both 10 and 20 kilometers from the beginning of the journey at the same time at time t.

At the same time, each function value can be achieved with several argument values.

So, we need to build a graph of the function, for this we need to make a table. Then study the function and its properties using the graph. But even before constructing a graph based on the type of function, we can say something about its properties: it is obvious that at cannot take negative values, since

So, let's make a table:

Rice. 1

From the graph it is easy to note the following properties:

Axis at- this is the axis of symmetry of the graph;

The vertex of the parabola is point (0; 0);

We see that the function only accepts non-negative values;

In the interval where the function decreases, and on the interval where the function increases;

The function acquires its smallest value at the vertex, ;

There is no greatest value of a function;

Example 1

Condition:

Solution:

Because the X by condition 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 of a function:

Solution:

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, and, therefore, on a given 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 function value can be achieved with several argument values.

A function graph is a visual representation of the behavior of a function on a coordinate plane. Graphs help you understand various aspects of a function that cannot be determined from the function itself. You can build graphs of many functions, and each of them will be given a specific formula. The graph of any function is built using a specific algorithm (if you have forgotten the exact process of graphing a specific function).

Steps

Graphing a Linear Function

Determine whether the function is linear. The linear function is given by a formula of the form F (x) = k x + b (\displaystyle F(x)=kx+b) or y = k x + b (\displaystyle y=kx+b)(for example, ), and its graph is a straight line. Thus, the formula includes one variable and one constant (constant) without any exponents, root signs, or the like. Given a function of a similar type, it is quite simple to plot a graph of such a function. Here are other examples of linear functions:

Use a constant to mark a point on the Y axis. The constant (b) is the “y” coordinate of the point where the graph intersects the Y axis. That is, it is a point whose “x” coordinate is equal to 0. Thus, if x = 0 is substituted into the formula, then y = b (constant). In our example y = 2 x + 5 (\displaystyle y=2x+5) the constant is equal to 5, that is, the point of intersection with the Y axis has coordinates (0.5). Plot this point on the coordinate plane.

Find the slope of the line. It is equal to the multiplier of the variable. In our example y = 2 x + 5 (\displaystyle y=2x+5) with the variable “x” there is a factor of 2; thus, the slope coefficient is equal to 2. The slope coefficient determines the angle of inclination of the straight line to the X axis, that is, the greater the slope coefficient, the faster the function increases or decreases.

Write the slope as a fraction. The angular coefficient is equal to the tangent of the angle of inclination, that is, the ratio of the vertical distance (between two points on a straight line) to the horizontal distance (between the same points). In our example, the slope is 2, so we can state that the vertical distance is 2 and the horizontal distance is 1. Write this as a fraction: 2 1 (\displaystyle (\frac (2)(1))).

• If the slope is negative, the function is decreasing.

1. From the point where the straight line intersects the Y axis, plot a second point using vertical and horizontal distances. A linear function can be graphed using two points. In our example, the intersection point with the Y axis has coordinates (0.5); From this point, move 2 spaces up and then 1 space to the right. Mark a point; it will have coordinates (1,7). Now you can draw a straight line.

   Using a ruler, draw a straight line through two points. To avoid mistakes, find the third point, but in most cases the graph can be plotted using two points. Thus, you have plotted a linear function.

   Plotting points on the coordinate plane

   1. Define a function. The function is denoted as f(x). All possible values ​​of the variable "y" are called the domain of the function, and all possible values ​​of the variable "x" are called the domain of the function. For example, consider the function y = x+2, namely f(x) = x+2.

      Draw two intersecting perpendicular lines. The horizontal line is the X axis. The vertical line is the Y axis.

      Label the coordinate axes. Divide each axis into equal segments and number them. The intersection point of the axes is 0. For the X axis: positive numbers are plotted to the right (from 0), and negative numbers to the left. For the Y axis: positive numbers are plotted on top (from 0), and negative numbers on the bottom.

      Find the values ​​of "y" from the values ​​of "x". In our example, f(x) = x+2. Substitute specific x values ​​into this formula to calculate the corresponding y values. If given a complex function, simplify it by isolating the “y” on one side of the equation.

      • -1: -1 + 2 = 1
      • 0: 0 +2 = 2
      • 1: 1 + 2 = 3

   2. Plot the points on the coordinate plane. For each pair of coordinates, do the following: find the corresponding value on the X axis and draw a vertical line (dotted); find the corresponding value on the Y axis and draw a horizontal line (dashed line). Mark the intersection point of the two dotted lines; thus, you have plotted a point on the graph.

      Erase the dotted lines. Do this after plotting all the points on the graph on the coordinate plane. Note: the graph of the function f(x) = x is a straight line passing through the coordinate center [point with coordinates (0,0)]; the graph f(x) = x + 2 is a line parallel to the line f(x) = x, but shifted upward by two units and therefore passing through the point with coordinates (0,2) (because the constant is 2).

   Graphing a Complex Function

   Find the zeros of the function. The zeros of a function are the values ​​of the x variable where y = 0, that is, these are the points where the graph intersects the X-axis. Keep in mind that not all functions have zeros, but they are the first step in the process of graphing any function. To find the zeros of a function, equate it to zero. For example:

   Find and mark the horizontal asymptotes. An asymptote is a line that the graph of a function approaches but never intersects (that is, in this region the function is not defined, for example, when dividing by 0). Mark the asymptote with a dotted line. If the variable "x" is in the denominator of a fraction (for example, y = 1 4 − x 2 (\displaystyle y=(\frac (1)(4-x^(2))))), set the denominator to zero and find “x”. In the obtained values ​​of the variable “x” the function is not defined (in our example, draw dotted lines through x = 2 and x = -2), because you cannot divide by 0. But asymptotes exist not only in cases where the function contains a fractional expression. Therefore, it is recommended to use common sense:

Constructing graphs of functions containing modules usually causes considerable difficulties for schoolchildren. However, everything is not so bad. It is enough to remember a few algorithms for solving such problems, and you can easily build a graph of even the most seemingly complex function. Let's figure out what kind of algorithms these are.

1. Plotting a graph of the function y = |f(x)|

Note that the set of function values ​​y = |f(x)| : y ≥ 0. Thus, the graphs of such functions are always located entirely in the upper half-plane.

Plotting a graph of the function y = |f(x)| consists of the following simple four steps.

1) Carefully and carefully construct a graph of the function y = f(x).

2) Leave unchanged all points on the graph that are above or on the 0x axis.

3) Display the part of the graph that lies below the 0x axis symmetrically relative to the 0x axis.

Example 1. Draw a graph of the function y = |x 2 – 4x + 3|

1) We build a graph of the function y = x 2 – 4x + 3. Obviously, the graph of this function is a parabola. Let's find the coordinates of all points of intersection of the parabola with the coordinate axes and the coordinates of the vertex of the parabola.

x 2 – 4x + 3 = 0.

x 1 = 3, x 2 = 1.

Therefore, the parabola intersects the 0x axis at points (3, 0) and (1, 0).

y = 0 2 – 4 0 + 3 = 3.

Therefore, the parabola intersects the 0y axis at the point (0, 3).

Parabola vertex coordinates:

x in = -(-4/2) = 2, y in = 2 2 – 4 2 + 3 = -1.

Therefore, point (2, -1) is the vertex of this parabola.

Draw a parabola using the data obtained (Fig. 1)

2) The part of the graph lying below the 0x axis is displayed symmetrically relative to the 0x axis.

3) We get a graph of the original function ( rice. 2, shown as a dotted line).

2. Graphing the function y = f(|x|)

Note that functions of the form y = f(|x|) are even:

y(-x) = f(|-x|) = f(|x|) = y(x). This means that the graphs of such functions are symmetrical about the 0y axis.

Plotting a graph of the function y = f(|x|) consists of the following simple chain of actions.

1) Graph the function y = f(x).

2) Leave that part of the graph for which x ≥ 0, that is, the part of the graph located in the right half-plane.

3) Display the part of the graph specified in point (2) symmetrically to the 0y axis.

4) As the final graph, select the union of the curves obtained in points (2) and (3).

Example 2. Draw a graph of the function y = x 2 – 4 · |x| + 3

Since x 2 = |x| 2, then the original function can be rewritten in the following form: y = |x| 2 – 4 |x| + 3. Now we can apply the algorithm proposed above.

1) We carefully and carefully build a graph of the function y = x 2 – 4 x + 3 (see also rice. 1).

2) We leave that part of the graph for which x ≥ 0, that is, the part of the graph located in the right half-plane.

3) Display the right side of the graph symmetrically to the 0y axis.

(Fig. 3).

Example 3. Draw a graph of the function y = log 2 |x|

We apply the scheme given above.

1) Build a graph of the function y = log 2 x (Fig. 4).

3. Plotting the function y = |f(|x|)|

Note that functions of the form y = |f(|x|)| are also even. Indeed, y(-x) = y = |f(|-x|)| = y = |f(|x|)| = y(x), and therefore, their graphs are symmetrical about the 0y axis. The set of values ​​of such functions: y ≥ 0. This means that the graphs of such functions are located entirely in the upper half-plane.

To plot the function y = |f(|x|)|, you need to:

1) Carefully construct a graph of the function y = f(|x|).

2) Leave unchanged the part of the graph that is above or on the 0x axis.

3) Display the part of the graph located below the 0x axis symmetrically relative to the 0x axis.

4) As the final graph, select the union of the curves obtained in points (2) and (3).

Example 4. Draw a graph of the function y = |-x 2 + 2|x| – 1|.

1) Note that x 2 = |x| 2. This means that instead of the original function y = -x 2 + 2|x| - 1

you can use the function y = -|x| 2 + 2|x| – 1, since their graphs coincide.

We build a graph y = -|x| 2 + 2|x| – 1. For this we use algorithm 2.

a) Graph the function y = -x 2 + 2x – 1 (Fig. 6).

b) We leave that part of the graph that is located in the right half-plane.

c) We display the resulting part of the graph symmetrically to the 0y axis.

d) The resulting graph is shown in the dotted line in the figure (Fig. 7).

2) There are no points above the 0x axis; we leave the points on the 0x axis unchanged.

3) The part of the graph located below the 0x axis is displayed symmetrically relative to 0x.

4) The resulting graph is shown in the figure with a dotted line (Fig. 8).

Example 5. Graph the function y = |(2|x| – 4) / (|x| + 3)|

1) First you need to plot the function y = (2|x| – 4) / (|x| + 3). To do this, we return to Algorithm 2.

a) Carefully plot the function y = (2x – 4) / (x + 3) (Fig. 9).

Note that this function is fractional linear and its graph is a hyperbola. To plot a curve, you first need to find the asymptotes of the graph. Horizontal – y = 2/1 (the ratio of the coefficients of x in the numerator and denominator of the fraction), vertical – x = -3.

2) We will leave that part of the graph that is above the 0x axis or on it unchanged.

3) The part of the graph located below the 0x axis will be displayed symmetrically relative to 0x.

4) The final graph is shown in the figure (Fig. 11).

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