Plot the function y 2 2x. Functions and Graphs

21.06.2020

Graphing functions is one of the features of Excel. In this article, we will look at the process of plotting some graphs mathematical functions: linear, quadratic and inverse proportionality.

A function is a set of points (x, y) that satisfies the expression y=f(x). Therefore, we need to fill in an array of such points, and Excel will build a function graph based on them.

1) Consider an example of plotting a graph of a linear function: y=5x-2

The graph of a linear function is a straight line that can be drawn from two points. Let's create a sign

In our case y=5x-2. To the cell with the first value y let's enter the formula: =5*D4-2. In another cell, the formula can be entered in the same way (by changing D4 on the D5) or use the autocomplete token.

As a result, we will get a table:

Now you can start creating a chart.

Choose: INSERT -> SPINT -> SPOT WITH SMOOTH CURVES AND MARKERS (I recommend using this particular type of chart)

An empty chart area will appear. Press the SELECT DATA button

Let's select the data: the range of cells of the abscissa axis (x) and the ordinate axis (y). As the name of the series, we can enter the function itself in quotes "y=5x-2" or something else. Here's what happened:

We press OK. Before us is a graph of a linear function.

2) Consider the process of plotting quadratic function- parabolas y=2x 2 -2

A parabola cannot be built from two points, unlike a straight line.

Let's set the spacing on the axis x on which our parabola will be built. I'll choose [-5; 5].

I'll take a step. How less step, the more accurate the graph will be. I will choose 0,2 .

Populate a column with values X, using the autocomplete token to the value x=5.

Value column at calculated by the formula: =2*B4^2-2. Using the autocomplete marker, we calculate the values at for others X.

Choose: INSERT -> POINT -> POINT WITH SMOOTH CURVES AND MARKERS and act in the same way as plotting a linear function graph.

To avoid dots on the chart, change the chart type to SPOT WITH SMOOTH CURVES.

Any other graphs of continuous functions are constructed in a similar way.

3) If the function is piecewise, then it is necessary to combine each “piece” of the graph in one area of the diagrams.

Let's look at this using the function as an example. y=1/x.

The function is defined on the intervals (- ins; 0) and (0; + ins)

Let's create a graph of the function on the intervals: [-4; 0) and (0; 4].

Let's prepare two tables, where x changes in steps 0,2 :

Find function values from each argument X similar to the examples above.

You must add two rows to the diagram - for the first and second plates, respectively.

We get the graph of the function y=1/x

In addition, I give a video - which shows the procedure described above.

In the next article I will tell you how to create 3-dimensional graphs in Excel.

Thank you for your attention!

The construction of graphs of functions containing modules usually causes considerable difficulties for schoolchildren. However, everything is not so bad. It is enough to remember several algorithms for solving such problems, and you can easily plot even the most seemingly complex function. Let's see what these algorithms are.

1. Plotting 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 completely in the upper half-plane.

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

1) Construct carefully and carefully the graph of the function y = f(x).

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

3) The part of the graph that lies below the 0x axis, display symmetrically about 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 \u003d x 2 - 4x + 3. It is obvious that 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 \u003d 0 2 - 4 0 + 3 \u003d 3.

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

Parabola vertex coordinates:

x in \u003d - (-4/2) \u003d 2, y in \u003d 2 2 - 4 2 + 3 \u003d -1.

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

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

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

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

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

1) Plot 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 paragraph (2) symmetrically to the 0y axis.

4) As the final graph, select the union of the curves obtained in paragraphs (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 as follows: y = |x| 2 – 4 · |x| + 3. And now we can apply the algorithm proposed above.

1) We build carefully and carefully the graph of the function y \u003d x 2 - 4 x + 3 (see also rice. one).

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) We plot 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. Hence, the graphs of such functions are located completely in the upper half-plane.

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

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

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

3) The part of the graph located below the 0x axis should be displayed symmetrically with respect to the 0x axis.

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

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

1) Note that x 2 = |x| 2. Hence, instead of the original function y = -x 2 + 2|x| - one

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

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

a) We plot the function y \u003d -x 2 + 2x - 1 (Fig. 6).

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

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

d) The resulting graph is shown in the figure with a dotted line (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 with respect to 0x.

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

Example 5. Plot 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 linear-fractional and its graph is a hyperbola. To build a curve, you first need to find the asymptotes of the graph. Horizontal - y \u003d 2/1 (the ratio of the coefficients at x in the numerator and denominator of a fraction), vertical - x \u003d -3.

2) The part of the chart that is above or on the 0x axis will be left unchanged.

3) The part of the chart located below the 0x axis will be displayed symmetrically with respect to 0x.

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

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The function graph is visual representation behavior of some function on the coordinate plane. Plots help to 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 by a specific formula. The graph of any function is built according to a certain algorithm (if you forgot the exact process of plotting a graph of a particular function).

Steps

Plotting a Linear Function

Determine if the function is linear. A 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, and the like. Given a function of a similar form, plotting such a function is quite simple. 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 intersection point of the graph with the Y-axis. That is, it is a point whose “x” coordinate is 0. Thus, if x = 0 is substituted into the formula, then y = b (constant). In our example y = 2x + 5 (\displaystyle y=2x+5) the constant is 5, that is, the point of intersection with the Y-axis has coordinates (0,5). Put this point on coordinate plane.

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

Write the slope as a fraction. The slope 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 say 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.

From the point where the line intersects with the Y axis, draw a second point using the vertical and horizontal distances. A linear function can be plotted using two points. In our example, the point of intersection 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.

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

Drawing 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 range 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. Break each axis into equal segments and number them. The intersection point of the axes is 0. For the X axis: positive numbers are plotted on the right (from 0), and negative numbers on the left. For the Y-axis: positive numbers are plotted on top (from 0), and negative numbers on the bottom.
  
  Find the "y" values from the "x" values. In our example f(x) = x+2. Substitute certain "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. Draw 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 line); find the corresponding value on the y-axis and draw a horizontal line (dotted line). Mark the point of intersection of the two dotted lines; thus, you have plotted a graph point.
  
  Erase the dotted lines. Do this after plotting all the graph points on the coordinate plane. Note: the graph of the function f(x) = x is a straight line passing through the center of coordinates [point with coordinates (0,0)]; the graph f(x) = x + 2 is a line parallel to the line f(x) = x, but shifted up by two units and therefore passing through the point with coordinates (0,2) (because the constant is 2).
Plotting a complex function

"Natural logarithm" - 0.1. natural logarithms. 4. "Logarithmic darts". 0.04. 7.121.

"Power function grade 9" - U. Cubic parabola. Y = x3. Grade 9 teacher Ladoshkina I.A. Y = x2. Hyperbola. 0. Y \u003d xn, y \u003d x-n where n is the given natural number. X. The exponent is an even natural number (2n).

"Quadratic function" - 1 Quadratic function definition 2 Function properties 3 Function graphs 4 Quadratic inequalities 5 Conclusion. Properties: Inequalities: Prepared by Andrey Gerlitz, a student of grade 8A. Plan: Graph: -Intervals of monotonicity at a > 0 at a< 0. Квадратичная функция. Квадратичные функции используются уже много лет.

"Quadratic function and its graph" - Decision. y \u003d 4x A (0.5: 1) 1 \u003d 1 A-belongs. When a=1, the formula y=ax takes the form.

"Class 8 quadratic function" - 1) Construct the top of the parabola. Plotting a quadratic function. x. -7. Plot the function. Algebra Grade 8 Teacher 496 school Bovina TV -1. Construction plan. 2) Construct the axis of symmetry x=-1. y.

Building charts online is a very useful way to graphically display something that cannot be expressed in words.

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This is where infographics come to the rescue, allowing you to present various kinds of information in a simple and expressive form.

However, the construction of infographic images requires a certain analytical thinking and a wealth of imagination.

We hasten to please you - there are enough resources on the Internet that provide online charting.

Yotx.ru

A wonderful Russian-language service that plots online graphs by points (by values) and graphs of functions (normal and parametric).

This site has an intuitive interface and is easy to use. It does not require registration, which significantly saves the user's time.

Allows you to quickly save ready-made graphics on your computer, and also generates code for posting on a blog or website.

Yotx.ru has a tutorial and chart examples that were created by users.

Perhaps, for people who study mathematics or physics in depth, this service will not be enough (for example, it is impossible to plot a graph in polar coordinates, since the service does not have a logarithmic scale), but it is quite enough to perform the simplest laboratory work.

The advantage of the service is that it does not force, like many other programs, to search for the result obtained over the entire two-dimensional plane.

The size of the graph and the intervals along the coordinate axes are automatically generated so that the graph is easy to view.

At the same time on the same plane it is possible to build several graphs.

Additionally, on the site you can use the matrix calculator, with which it is easy to perform various actions and transformations.

ChartGo

English-language service for the development of multifunctional and multi-colored histograms, line charts, pie charts.

A detailed manual and demo videos are presented to users for training.

ChartGo will be useful for those who need it regularly. Among similar resources, “Create a graph online quickly” is distinguished by its simplicity.

Charting online is carried out according to the table.

At the beginning of the work, you must select one of the types of charts.

The application provides users with a number of simple charting customization options various functions in 2D and 3D coordinates.

You can select one of the chart types and switch between 2D and 3D.

Size settings provide maximum control between vertical and horizontal orientation.

Users can customize their charts with a unique title, as well as name the X and Y elements.

To plot online xyz graphs in the "Example" section, many layouts are available that you can change to your liking.

Note! In ChartGo, many charts can be built in one rectangular system. Each graph is made up of points and lines. Functions of a real variable (analytical) are set by the user in a parametric form.

Additional functionality has also been developed, which includes monitoring and displaying coordinates on a plane or in a three-dimensional system, importing and exporting numerical data in certain formats.

The program has a highly customizable interface.

After creating a diagram, the user can use the function to print the result and save the graph as a static picture.

OnlineCharts.ru

You can find another great application for a spectacular presentation of information on the OnlineCharts.ru website, where you can plot a function graph online for free.

The service is able to work with many types of charts, including line, bubble, pie, column and radial.

The system has a very simple and intuitive interface. All available functions are separated by tabs in the form of a horizontal menu.

To get started, you need to select the type of chart you want to build.

After that, you can configure some additional options appearance, depending on the selected chart type.

In the "Add data" tab, the user is prompted to set the number of rows and, if necessary, the number of groups.

You can also define a color.

Note! The “Signatures and fonts” tab offers to set the properties of the signatures (should they be displayed at all, if so, what color and font size). It also provides the ability to select the font type and size for the main text of the chart.

Everything is extremely simple.

Aiportal.ru

The simplest and least functional of all the online services presented here. It will not be possible to create a three-dimensional graph online on this site.

It is designed to plot complex functions in a coordinate system at a certain range of values.

For the convenience of users, the service provides reference data on the syntax of various mathematical operations, as well as on the list of supported functions and constant values.

All data necessary for drawing up the schedule is entered into the "Functions" window. At the same time, the user can build several graphs on the same plane.

Therefore, it is allowed to add several functions in a row, but after each function, you must insert a semicolon. The construction area is also set.

It is possible to build graphs online according to the table or without it. Color legend supported.

Despite the poor functionality, it is still an online service, so you do not have to search, download and install any software for a long time.

To build a graph, you just need to have it from any available device: PC, laptop, tablet or smartphone.

Plotting a function online

TOP 4 best online charting services