Graph of the function e to the power. Functions

The graph of a function is visual representation behavior of some function on the 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. If a function of a similar type is given, 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). Place this point on 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:

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 it manually or using 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.

One of the most famous exponential functions in mathematics is an exponent. It represents the Euler number raised to the specified power. In Excel there is a separate operator that allows you to calculate it. Let's see how it can be used in practice.

The exponent is the Euler number raised to a given power. The Euler number itself is approximately 2.718281828. Sometimes it is also called the Napier number. The exponent function looks like this:

where e is the Euler number and n is the degree of raising.

To calculate this indicator in Excel, a separate operator is used - EXP. In addition, this function can be displayed as a graph. We will talk about working with these tools further.

Method 1: Calculate the exponent by manually entering the function

EXP(number)

That is, this formula contains only one argument. It is precisely the power to which the Euler number must be raised. This argument could be of the form numerical value, and take the form of a reference to a cell containing an exponent.

Method 2: Using the Function Wizard

Although the syntax for calculating the exponent is extremely simple, some users prefer to use Function Wizard. Let's look at how this is done with an example.

If a cell reference that contains an exponent is used as an argument, then you need to place the cursor in the field "Number" and simply select that cell on the sheet. Its coordinates will immediately be displayed in the field. After this, to calculate the result, click on the button "OK".

Method 3: plotting

In addition, in Excel it is possible to construct a graph using the results obtained from calculating the exponent as a basis. To construct a graph, the sheet must already have calculated values ​​of the exponent of various powers. They can be calculated using one of the methods described above.