Schoolchildren are faced with the task of constructing a function graph at the very beginning of studying algebra and continue to build them from year to year. Starting from a graph of a linear function, for the construction of which you need to know only two points, to a parabola, for which you already need 6 points, a hyperbola and a sinusoid. Every year, the functions become more and more complex and it is no longer possible to plot their graphs according to a template, it is necessary to conduct more complex studies using derivatives and limits.

Let's figure out how to find the graph of a function? To do this, let's start with the most simple functions, whose graphs are built by points, and then we will consider a plan for constructing more complex functions.

Plotting a Linear Function

To build the simplest graphs, a table of function values ​​is used. The graph of a linear function is a straight line. Let's try to find the points of the graph of the function y=4x+5.

1. To do this, we take two arbitrary values ​​of the variable x, substitute them one by one into the function, find the value of the variable y and put everything in the table.
2. Let's take the value x=0 and substitute it into the function instead of x - 0. We get: y=4*0+5, that is, y=5 write this value to the table under 0. Similarly, take x=0 we get y=4*1+5 , y=9.
3. Now, to build a function graph, you need to plot these points on the coordinate plane. Then you need to draw a straight line.

Plotting a Quadratic Function

A quadratic function is a function of the form y=ax 2 +bx +c, where x is a variable, a,b,c are numbers (a is not equal to 0). For example: y=x 2 , y=x 2 +5, y=(x-3) 2 , y=2x 2 +3x+5.

To build the simplest quadratic function y=x 2 usually take 5-7 points. Let's take the values ​​for the variable x: -2, -1, 0, 1, 2 and find the values ​​of y in the same way as when building the first graph.

The graph of a quadratic function is called a parabola. After constructing function graphs, students have new tasks associated with the graph.

Example 1: find the abscissa of the function graph point y=x 2 if the ordinate is 9. To solve the problem, you need to substitute its value 9 instead of y into the function. We get 9=x 2 and solve this equation. x=3 and x=-3. This can also be seen in the graph of the function.

Investigation of a function and construction of its graph

To plot more complex functions, you need to perform several steps aimed at its study. For this you need:

1. Find the scope of the function. The scope is all the values ​​that x can take on. From the domain of definition, one should exclude those points at which the denominator becomes 0 or the radical expression becomes negative.
2. Set even or odd function. Recall that even is the function that meets the condition f(-x)=f(x). Its graph is symmetrical with respect to Oy. A function will be odd if it meets the condition f(-x)=-f(x). In this case, the graph is symmetrical about the origin.
3. Find points of intersection with coordinate axes. In order to find the abscissa of the point of intersection with the x-axis, it is necessary to solve the equation f(x)=0 (the ordinate is 0). To find the ordinate of the point of intersection with the Oy axis, it is necessary to substitute 0 in the function instead of the x variable (the abscissa is 0).
4. Find the asymptotes of the function. An asymptote is a line that the graph approaches indefinitely, but never crosses. Let's figure out how to find the asymptotes of the graph of a function.
   • Vertical asymptote straight line of the form x=a
   • Horizontal asymptote - a straight line of the form y \u003d a
   • Oblique asymptote - straight line of the form y=kx+b
5. Find the extremum points of the function, the intervals of increase and decrease of the function. Find the extremum points of the function. To do this, you need to find the first derivative and equate it to 0. It is at these points that the function can change from increasing to decreasing. Let us determine the sign of the derivative on each interval. If the derivative is positive, then the graph of the function increases; if it is negative, it decreases.
6. Find the inflection points of the graph of the function, the intervals of convexity up and down.

Finding inflection points is now easier than ever. You just need to find the second derivative, then equate it to zero. Next, we find the sign of the second derivative on each interval. If positive, then the graph of the function is convex down, if negative - up.

A linear function is a function of the form y=kx+b, where x is an independent variable, k and b are any numbers.
The graph of a linear function is a straight line.

1. To plot a function graph, we need the coordinates of two points belonging to the graph of the function. To find them, you need to take two x values, substitute them into the equation of the function, and calculate the corresponding y values ​​from them.

For example, to plot the function y= x+2, it is convenient to take x=0 and x=3, then the ordinates of these points will be equal to y=2 and y=3. We get points A(0;2) and B(3;3). Let's connect them and get the graph of the function y= x+2:

2. In the formula y=kx+b, the number k is called the proportionality factor:
if k>0, then the function y=kx+b increases
if k
The coefficient b shows the shift of the graph of the function along the OY axis:
if b>0, then the graph of the function y=kx+b is obtained from the graph of the function y=kx by shifting b units up along the OY axis
if b
The figure below shows the graphs of the functions y=2x+3; y= ½x+3; y=x+3

Note that in all these functions the coefficient k Above zero, and functions are increasing. Moreover, the greater the value of k, the greater the angle of inclination of the straight line to the positive direction of the OX axis.

In all functions b=3 - and we see that all graphs intersect the OY axis at the point (0;3)

Now consider the graphs of functions y=-2x+3; y=- ½ x+3; y=-x+3

This time, in all functions, the coefficient k less than zero and features decrease. The coefficient b=3, and the graphs, as in the previous case, cross the OY axis at the point (0;3)

Consider the graphs of functions y=2x+3; y=2x; y=2x-3

Now, in all equations of functions, the coefficients k are equal to 2. And we got three parallel lines.

But the coefficients b are different, and these graphs intersect the OY axis at different points:
The graph of the function y=2x+3 (b=3) crosses the OY axis at the point (0;3)
The graph of the function y=2x (b=0) crosses the OY axis at the point (0;0) - the origin.
The graph of the function y=2x-3 (b=-3) crosses the OY axis at the point (0;-3)

So, if we know the signs of the coefficients k and b, then we can immediately imagine what the graph of the function y=kx+b looks like.
If a k 0

If a k>0 and b>0, then the graph of the function y=kx+b looks like:

If a k>0 and b, then the graph of the function y=kx+b looks like:

If a k, then the graph of the function y=kx+b looks like:

If a k=0, then the function y=kx+b turns into a function y=b and its graph looks like:

The ordinates of all points of the graph of the function y=b are equal to b If b=0, then the graph of the function y=kx (direct proportionality) passes through the origin:

3. Separately, we note the graph of the equation x=a. The graph of this equation is a straight line parallel to the OY axis, all points of which have an abscissa x=a.

For example, the graph of the equation x=3 looks like this:
Attention! The equation x=a is not a function, so one value of the argument corresponds to different meanings function, which does not match the function definition.

4. Condition for parallelism of two lines:

The graph of the function y=k 1 x+b 1 is parallel to the graph of the function y=k 2 x+b 2 if k 1 =k 2

5. The condition for two straight lines to be perpendicular:

The graph of the function y=k 1 x+b 1 is perpendicular to the graph of the function y=k 2 x+b 2 if k 1 *k 2 =-1 or k 1 =-1/k 2

6. Intersection points of the graph of the function y=kx+b with the coordinate axes.

with OY axis. The abscissa of any point belonging to the OY axis is equal to zero. Therefore, to find the point of intersection with the OY axis, you need to substitute zero instead of x in the equation of the function. We get y=b. That is, the point of intersection with the OY axis has coordinates (0;b).

With the x-axis: The ordinate of any point belonging to the x-axis is zero. Therefore, to find the point of intersection with the OX axis, you need to substitute zero instead of y in the equation of the function. We get 0=kx+b. Hence x=-b/k. That is, the point of intersection with the OX axis has coordinates (-b / k; 0):

A function graph is a visual representation of the 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). 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 = 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.

1. 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

   Find the zeros of the function. The zeros of a function are the values ​​of the variable "x" at which y = 0, that is, these are the points of intersection of the graph with the x-axis. Keep in mind that not all functions have zeros, but this is the first step in the process of plotting a graph of any function. To find the zeros of a function, set it equal to zero. For example:

   Find and label the horizontal asymptotes. An asymptote is a line that the graph of a function approaches but never crosses (that is, the function is not defined in this area, for example, when divided 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 dashed 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:

First, try to find the scope of the function:

Did you manage? Let's compare the answers:

All right? Well done!

Now let's try to find the range of the function:

Found? Compare:

Did it agree? Well done!

Let's work with the graphs again, only now it's a little more difficult - to find both the domain of the function and the range of the function.

How to Find Both the Domain and Range of a Function (Advanced)

Here's what happened:

With graphics, I think you figured it out. Now let's try to find the domain of the function in accordance with the formulas (if you don't know how to do this, read the section about):

Did you manage? Checking answers:

1. , since the root expression must be greater than or equal to zero.
2. , since it is impossible to divide by zero and the radical expression cannot be negative.
3. , since, respectively, for all.
4. because you can't divide by zero.

However, we still have one more moment that has not been sorted out ...

Let me reiterate the definition and focus on it:

Noticed? The word "only" is a very, very important element of our definition. I will try to explain to you on the fingers.

Let's say we have a function given by a straight line. . When, we substitute this value into our "rule" and get that. One value corresponds to one value. We can even make a table of various values ​​and plot a given function to verify this.

"Look! - you say, - "" meets twice!" So maybe the parabola is not a function? No, it is!

The fact that "" occurs twice is far from a reason to accuse the parabola of ambiguity!

The fact is that, when calculating for, we got one game. And when calculating with, we got one game. So that's right, the parabola is a function. Look at the chart:

Got it? If not, here's a real-life example for you, far from mathematics!

Let's say we have a group of applicants who met when submitting documents, each of whom told in a conversation where he lives:

Agree, it is quite realistic that several guys live in the same city, but it is impossible for one person to live in several cities at the same time. This is, as it were, a logical representation of our "parabola" - Several different x's correspond to the same y.

Now let's come up with an example where the dependency is not a function. Let's say these same guys told what specialties they applied for:

Here we have a completely different situation: one person can easily apply for one or several directions. That is one element sets are put in correspondence multiple elements sets. Respectively, it's not a function.

Let's test your knowledge in practice.

Determine from the pictures what is a function and what is not:

Got it? And here is answers:

• The function is - B,E.
• Not a function - A, B, D, D.

You ask why? Yes, here's why:

In all figures except AT) and E) there are several for one!

I am sure that now you can easily distinguish a function from a non-function, say what an argument is and what a dependent variable is, and also determine the scope of the argument and the scope of the function. Let's move on to the next section - how to define a function?

Ways to set a function

What do you think the words mean "set function"? That's right, it means explaining to everyone what function we are talking about in this case. Moreover, explain in such a way that everyone understands you correctly and the graphs of functions drawn by people according to your explanation were the same.

How can I do that? How to set a function? The easiest way, which has already been used more than once in this article - using a formula. We write a formula, and by substituting a value into it, we calculate the value. And as you remember, a formula is a law, a rule according to which it becomes clear to us and to another person how an X turns into a Y.

Usually, this is exactly what they do - in tasks we see ready-made functions defined by formulas, however, there are other ways to set a function that everyone forgets about, and therefore the question “how else can you set a function?” confuses. Let's take a look at everything in order, and start with the analytical method.

Analytical way of defining a function

The analytical method is the task of a function using a formula. This is the most universal and comprehensive and unambiguous way. If you have a formula, then you know absolutely everything about the function - you can make a table of values ​​​​on it, you can build a graph, determine where the function increases and where it decreases, in general, explore it in full.

Let's consider a function. What does it matter?

"What does it mean?" - you ask. I'll explain now.

Let me remind you that in the notation, the expression in brackets is called the argument. And this argument can be any expression, not necessarily simple. Accordingly, whatever the argument (expression in brackets), we will write it instead in the expression.

In our example, it will look like this:

Consider another task related to the analytical method of specifying a function that you will have on the exam.

Find the value of the expression, at.

I'm sure that at first, you were scared when you saw such an expression, but there is absolutely nothing scary in it!

Everything is the same as in the previous example: whatever the argument (expression in brackets), we will write it instead in the expression. For example, for a function.

What should be done in our example? Instead, you need to write, and instead of -:

shorten the resulting expression:

That's all!

Independent work

Now try to find the meaning of the following expressions yourself:

1. , if
2. , if

Did you manage? Let's compare our answers: We are used to the fact that the function has the form

Even in our examples, we define the function in this way, but analytically it is possible to define the function implicitly, for example.

Try building this function yourself.

Did you manage?

Here's how I built it.

What equation did we end up with?

Correctly! Linear, which means that the graph will be a straight line. Let's make a table to determine which points belong to our line:

That's just what we were talking about ... One corresponds to several.

Let's try to draw what happened:

Is what we got a function?

That's right, no! Why? Try to answer this question with a picture. What did you get?

“Because one value corresponds to several values!”

What conclusion can we draw from this?

That's right, a function can't always be expressed explicitly, and what's "disguised" as a function isn't always a function!

Tabular way of defining a function

As the name suggests, this method is a simple plate. Yes Yes. Like the one we already made. For example:

Here you immediately noticed a pattern - Y is three times larger than X. And now the “think very well” task: do you think that a function given in the form of a table is equivalent to a function?

Let's not talk for a long time, but let's draw!

So. We draw a function given in both ways:

Do you see the difference? It's not about the marked points! Take a closer look:

Have you seen it now? When we set the function in a tabular way, we reflect on the graph only those points that we have in the table and the line (as in our case) passes only through them. When we define a function in an analytical way, we can take any points, and our function is not limited to them. Here is such a feature. Remember!

Graphical way to build a function

The graphical way of constructing a function is no less convenient. We draw our function, and another interested person can find what y is equal to at a certain x, and so on. Graphical and analytical methods are among the most common.

However, here you need to remember what we talked about at the very beginning - not every “squiggle” drawn in the coordinate system is a function! Remembered? Just in case, I'll copy here the definition of what a function is:

As a rule, people usually name exactly those three ways of specifying a function that we have analyzed - analytical (using a formula), tabular and graphic, completely forgetting that a function can be described verbally. Like this? Yes, very easy!

Verbal description of the function

How to describe the function verbally? Let's take our recent example - . This function can be described as "each real value of x corresponds to its triple value." That's all. Nothing complicated. Of course, you will object - “there are such complex functions that it is simply impossible to set verbally!” Yes, there are some, but there are functions that are easier to describe verbally than to set with a formula. For example: "each natural value of x corresponds to the difference between the digits of which it consists, while the largest digit contained in the number entry is taken as the minuend." Now consider how our verbal description of the function is implemented in practice:

The largest digit in a given number -, respectively, - is reduced, then:

Main types of functions

Now let's move on to the most interesting - consider the main types of functions with which you worked / work and will work in the course of school and institute mathematics, that is, we will get to know them, so to speak, and give them brief description. Read more about each function in the corresponding section.

Linear function

A function of the form, where, are real numbers.

The graph of this function is a straight line, so the construction of a linear function is reduced to finding the coordinates of two points.

The position of the straight line on the coordinate plane depends on the slope.

Function scope (aka argument range) - .

The range of values ​​is .

quadratic function

Function of the form, where

The graph of the function is a parabola, when the branches of the parabola are directed downwards, when - upwards.

Many properties of a quadratic function depend on the value of the discriminant. The discriminant is calculated by the formula

The position of the parabola on the coordinate plane relative to the value and coefficient is shown in the figure:

Domain

The range of values ​​depends on the extremum of the given function (the vertex of the parabola) and the coefficient (the direction of the branches of the parabola)

Inverse proportionality

The function given by the formula, where

The number is called the inverse proportionality factor. Depending on what value, the branches of the hyperbola are in different squares:

Domain - .

The range of values ​​is .

SUMMARY AND BASIC FORMULA

1. A function is a rule according to which each element of a set is assigned a unique element of the set.

• - this is a formula denoting a function, that is, the dependence of one variable on another;
• - variable, or, argument;
• - dependent value - changes when the argument changes, that is, according to some specific formula that reflects the dependence of one value on another.

2. Valid argument values, or the scope of a function, is what is related to the possible under which the function makes sense.

3. Range of function values- this is what values ​​it takes, with valid values.

4. There are 4 ways to set the function:

• analytical (using formulas);
• tabular;
• graphic
• verbal description.

5. Main types of functions:

• : , where, are real numbers;
• : , where;
• : , where.

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 best browser for working with charts on this page of the site is Google Chrome. When using other browsers, correct operation is not guaranteed.