Graphs of functions and their formulas how to solve. Basic elementary functions: their properties and graphs
Knowledge basic elementary functions, their properties and graphs no less important than knowing the multiplication table. They are like a foundation, everything is based on them, everything is built from them, and everything comes down to them.
In this article, we list all the main elementary functions, give their graphs and give them without derivation and proofs. properties of basic elementary functions according to the scheme:
- behavior of the function on the boundaries of the domain of definition, vertical asymptotes (if necessary, see the article classification of breakpoints of a function);
- even and odd;
- convexity (convexity upwards) and concavity (convexity downwards) intervals, inflection points (if necessary, see the article function convexity, convexity direction, inflection points, convexity and inflection conditions);
- oblique and horizontal asymptotes;
- singular points of functions;
- special properties of some functions (for example, the smallest positive period for trigonometric functions).
If you are interested in or, then you can go to these sections of the theory.
Basic elementary functions are: constant function (constant), root of the nth degree, power function, exponential, logarithmic function, trigonometric and inverse trigonometric functions.
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Permanent function.
constant function is given on the set of all real numbers by the formula , where C is some real number. The constant function assigns to each real value of the independent variable x the same value of the dependent variable y - the value С. A constant function is also called a constant.
The graph of a constant function is a straight line parallel to the x-axis and passing through a point with coordinates (0,C) . For example, let's show graphs of constant functions y=5 , y=-2 and , which in the figure below correspond to the black, red and blue lines, respectively.
Properties of a constant function.
- Domain of definition: the whole set of real numbers.
- The constant function is even.
- Range of values: set consisting of a single number C .
- A constant function is non-increasing and non-decreasing (that's why it is constant).
- It makes no sense to talk about the convexity and concavity of the constant.
- There is no asymptote.
- The function passes through the point (0,C) coordinate plane.
The root of the nth degree.
Consider the basic elementary function, which is given by the formula , where n is natural number, greater than one.
The root of the nth degree, n is an even number.
Let's start with the nth root function for even values of the root exponent n .
For example, we give a picture with images of graphs of functions and , they correspond to black, red and blue lines.
The graphs of the functions of the root of an even degree have a similar form for other values of the indicator.
Properties of the root of the nth degree for even n .
The root of the nth degree, n is an odd number.
The root function of the nth degree with an odd exponent of the root n is defined on the entire set of real numbers. For example, we present graphs of functions and , the black, red, and blue curves correspond to them.
For other odd values of the root exponent, the graphs of the function will have a similar appearance.
Properties of the root of the nth degree for odd n .
Power function.
The power function is given by a formula of the form .
Consider the type of graphs of a power function and the properties of a power function depending on the value of the exponent.
Let's start with a power function with an integer exponent a . In this case, the form of graphs of power functions and the properties of functions depend on the even or odd exponent, as well as on its sign. Therefore, we first consider power functions for odd positive values of the exponent a , then for even positive ones, then for odd negative exponents, and finally, for even negative a .
The properties of power functions with fractional and irrational exponents (as well as the type of graphs of such power functions) depend on the value of the exponent a. We will consider them, firstly, when a is from zero to one, secondly, when a is greater than one, thirdly, when a is from minus one to zero, and fourthly, when a is less than minus one.
In conclusion of this subsection, for the sake of completeness, we describe a power function with zero exponent.
Power function with odd positive exponent.
Consider a power function with an odd positive exponent, that is, with a=1,3,5,… .
The figure below shows graphs of power functions - black line, - blue line, - red line, - green line. For a=1 we have linear function y=x .
Properties of a power function with an odd positive exponent.
Power function with even positive exponent.
Consider a power function with an even positive exponent, that is, for a=2,4,6,… .
As an example, let's take graphs of power functions - black line, - blue line, - red line. For a=2 we have a quadratic function whose graph is quadratic parabola.
Properties of a power function with an even positive exponent.
Power function with an odd negative exponent.
Look at the graphs of the exponential function for odd negative values of the exponent, that is, for a \u003d -1, -3, -5, ....
The figure shows graphs of exponential functions as examples - black line, - blue line, - red line, - green line. For a=-1 we have inverse proportionality, whose graph is hyperbola.
Properties of a power function with an odd negative exponent.
Power function with an even negative exponent.
Let's move on to the power function at a=-2,-4,-6,….
The figure shows graphs of power functions - black line, - blue line, - red line.
Properties of a power function with an even negative exponent.
A power function with a rational or irrational exponent whose value is greater than zero and less than one.
Note! If a is a positive fraction with an odd denominator, then some authors consider the interval to be the domain of the power function. At the same time, it is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and the beginnings of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values of the argument. We will adhere to just such a view, that is, we will consider the domains of power functions with fractional positive exponents to be the set . We encourage students to get your teacher's perspective on this subtle point to avoid disagreement.
Consider a power function with rational or irrational exponent a , and .
We present graphs of power functions for a=11/12 (black line), a=5/7 (red line), (blue line), a=2/5 (green line).
A power function with a non-integer rational or irrational exponent greater than one.
Consider a power function with a non-integer rational or irrational exponent a , and .
Let us present the graphs of the power functions given by the formulas (black, red, blue and green lines respectively).
>For other values of the exponent a , the graphs of the function will have a similar look.
Power function properties for .
A power function with a real exponent that is greater than minus one and less than zero.
Note! If a is a negative fraction with an odd denominator, then some authors consider the interval . At the same time, it is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and the beginnings of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values of the argument. We will adhere to just such a view, that is, we will consider the domains of power functions with fractional negative exponents to be the set, respectively. We encourage students to get your teacher's perspective on this subtle point to avoid disagreement.
We pass to the power function , where .
In order to have a good idea of the type of graphs of power functions for , we give examples of graphs of functions (black, red, blue, and green curves, respectively).
Properties of a power function with exponent a , .
A power function with a non-integer real exponent that is less than minus one.
Let us give examples of graphs of power functions for , they are depicted in black, red, blue and green lines, respectively.
Properties of a power function with a non-integer negative exponent less than minus one.
When a=0 and we have a function - this is a straight line from which the point (0; 1) is excluded (the expression 0 0 was agreed not to attach any importance).
Exponential function.
One of the basic elementary functions is the exponential function.
Schedule exponential function, where and takes different kind depending on the value of the base a. Let's figure it out.
First, consider the case when the base of the exponential function takes a value from zero to one, that is, .
For example, we present the graphs of the exponential function for a = 1/2 - the blue line, a = 5/6 - the red line. The graphs of the exponential function have a similar appearance for other values of the base from the interval .
Properties of an exponential function with a base less than one.
We turn to the case when the base of the exponential function is greater than one, that is, .
As an illustration, we present graphs of exponential functions - the blue line and - the red line. For other values of the base, greater than one, the graphs of the exponential function will have a similar appearance.
Properties of an exponential function with a base greater than one.
Logarithmic function.
The next basic elementary function is the logarithmic function , where , . The logarithmic function is defined only for positive values of the argument, that is, for .
The graph of the logarithmic function takes on a different form depending on the value of the base a.
Let's start with the case when .
For example, we present the graphs of the logarithmic function for a = 1/2 - the blue line, a = 5/6 - the red line. For other values of the base, not exceeding one, the graphs of the logarithmic function will have a similar appearance.
Properties of a logarithmic function with a base less than one.
Let's move on to the case when the base of the logarithmic function is greater than one ().
Let's show graphs of logarithmic functions - blue line, - red line. For other values of the base, greater than one, the graphs of the logarithmic function will have a similar appearance.
Properties of a logarithmic function with a base greater than one.
Trigonometric functions, their properties and graphs.
All trigonometric functions (sine, cosine, tangent and cotangent) are basic elementary functions. Now we will consider their graphs and list their properties.
Trigonometric functions have the concept periodicity(recurrence of function values for different values of the argument that differ from each other by the value of the period , where T is the period), therefore, an item has been added to the list of properties of trigonometric functions "smallest positive period". Also, for each trigonometric function, we will indicate the values of the argument at which the corresponding function vanishes.
Now let's deal with all the trigonometric functions in order.
The sine function y = sin(x) .
Let's portray function graph sine, it is called "sine wave".
Properties of the sine function y = sinx .
Cosine function y = cos(x) .
The graph of the cosine function (it is called "cosine") looks like this:
Cosine function properties y = cosx .
Tangent function y = tg(x) .
The graph of the tangent function (it is called the "tangentoid") looks like:
Function properties tangent y = tgx .
Cotangent function y = ctg(x) .
Let's draw a graph of the cotangent function (it's called a "cotangentoid"):
Cotangent function properties y = ctgx .
Inverse trigonometric functions, their properties and graphs.
The inverse trigonometric functions (arcsine, arccosine, arctangent and arccotangent) are the basic elementary functions. Often, because of the prefix "arc", inverse trigonometric functions are called arc functions. Now we will consider their graphs and list their properties.
Arcsine function y = arcsin(x) .
Let's plot the arcsine function:
Function properties arccotangent y = arcctg(x) .Bibliography.
- Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. Algebra and the Beginnings of Analysis: Proc. for 10-11 cells. educational institutions.
- Vygodsky M.Ya. Handbook of elementary mathematics.
- Novoselov S.I. Algebra and elementary functions.
- Tumanov S.I. Elementary Algebra. A guide for self-education.
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 coefficient:
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 k 0
If k>0 and b>0, then the graph of the function y=kx+b looks like:
If k>0 and b, then the graph of the function y=kx+b looks like:
If k, then the graph of the function y=kx+b looks like:
If 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, since one value of the argument corresponds to different values of the function, which does not correspond to the definition of the function.
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):
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:
- , since the root expression must be greater than or equal to zero.
- , since it is impossible to divide by zero and the radical expression cannot be negative.
- , since, respectively, for all.
- 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 different meanings and build a graph of this function to make sure of 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 life example far from math!
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 IN) 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 in this case in question. 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:
- , If
- , 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?
Right! 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 figure in given number- , respectively, - 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
Function of the form where, - 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 values argument, 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.
Functions and their graphs are one of the most fascinating topics in school mathematics. It's just a pity that she passes... past the lessons and past the students. There is never enough time for her in high school. And those functions that take place in the 7th grade - a linear function and a parabola - are too simple and uncomplicated to show all the variety of interesting tasks.
The ability to build graphs of functions is necessary for solving problems with parameters on the exam in mathematics. This is one of the first topics of the course. mathematical analysis at the university. This is such an important topic that we, at the Unified State Exam-Studio, conduct special intensive courses on it for high school students and teachers, in Moscow and online. And often the participants say: “It is a pity that we did not know this before.”
But that's not all. It is with the concept of a function that real, “adult” mathematics begins. After all, addition and subtraction, multiplication and division, fractions and proportions - this is still arithmetic. Expression transformations are algebra. And mathematics is a science not only about numbers, but also about the relationships of quantities. The language of functions and graphs is understandable to a physicist, a biologist, and an economist. And as Galileo Galilei said, "The book of nature is written in the language of mathematics".
More precisely, Galileo Galilei said this: “Mathematics is the alphabet by which the Lord drew the Universe.”
Topics to review:
1. Graph the function
A familiar challenge! These met in OGE options mathematics. There they were considered difficult. But there is nothing complicated here.
Let's simplify the function formula:
Function graph - straight line with a punched out point
2. Graph the function
Let's select the integer part in the function formula:
The graph of the function is a hyperbola shifted 3 to the right in x and 2 up in y and stretched 10 times compared to the function graph
The selection of the integer part is a useful technique used in solving inequalities, plotting graphs and estimating integers in problems on numbers and their properties. You will also meet him in the first year, when you have to take integrals.
3. Graph the function
It is obtained from the graph of the function by stretching 2 times, flipping vertically and shifting 1 up vertically
4. Graph the function
The main thing is the correct sequence of actions. Let's write the function formula in a more convenient form:
We act in order:
1) Shift the graph of the function y=sinx to the left;
2) squeeze 2 times horizontally,
3) stretch 3 times vertically,
4) move up by 1
Now we will build several graphs of fractional rational functions. To better understand how we do this, read the article “Function Behavior at Infinity. Asymptotes".
5. Graph the function
Function scope:
Function zeros: and
The straight line x = 0 (y-axis) is the vertical asymptote of the function. Asymptote- a straight line, to which the graph of a function approaches infinitely close, but does not intersect it and does not merge with it (see the topic "Behavior of a function at infinity. Asymptotes")
Are there other asymptotes for our function? To find out, let's see how the function behaves as x goes to infinity.
Let's open the brackets in the function formula:
If x goes to infinity, then it goes to zero. The straight line is an oblique asymptote to the graph of the function.
6. Graph the function
This is a fractional rational function.
Function scope
Function zeros: points - 3, 2, 6.
The intervals of sign constancy of the function will be determined using the method of intervals.
Vertical asymptotes:
If x tends to infinity, then y tends to 1. Hence, is a horizontal asymptote.
Here is a sketch of the graph:
Another interesting technique is the addition of graphs.
7. Graph the function
If x tends to infinity, then the graph of the function will approach infinitely close to the oblique asymptote
If x tends to zero, then the function behaves like This is what we see on the graph:
So we have built a graph of the sum of functions. Now the work schedule!
8. Graph the function
The scope of this function is positive numbers, because only for positive x is defined
The function values are zero at (when the logarithm is zero), as well as at points where, that is, at
When , the value (cos x) is equal to one. The value of the function at these points will be equal to
9. Graph the function
The function is defined for It is even, since it is the product of two odd functions and The graph is symmetrical about the y-axis.
The zeros of the function are at points where, that is, at
If x goes to infinity, goes to zero. But what happens if x tends to zero? After all, both x and sin x will become smaller and smaller. How will the private behave?
It turns out that if x tends to zero, then it tends to one. In mathematics, this statement is called the "First Remarkable Limit."
But what about the derivative? Yes, we finally got there. The derivative helps to plot functions more accurately. Find maximum and minimum points, as well as function values at these points.
10. Graph the function
The scope of the function is all real numbers, since
The function is odd. Its graph is symmetrical with respect to the origin.
At x=0 the value of the function is equal to zero. For the values of the function are positive, for are negative.
If x goes to infinity, then it goes to zero.
Let's find the derivative of the function
According to the formula for the derivative of a quotient,
If or
At the point, the derivative changes sign from "minus" to "plus", - the minimum point of the function.
At the point, the derivative changes sign from "plus" to "minus", - the maximum point of the function.
Let's find the values of the function at x=2 and at x=-2.
It is convenient to build function graphs according to a certain algorithm, or scheme. Remember you studied it in school?
The general scheme for constructing a graph of a function:
1. Function scope
2. Range of function values
3. Even - odd (if any)
4. Frequency (if any)
5. Zeros of the function (points where the graph crosses the coordinate axes)
6. Intervals of constancy of a function (that is, intervals on which it is strictly positive or strictly negative).
7. Asymptotes (if any).
8. Behavior of a function at infinity
9. Derivative of a function
10. Intervals of increase and decrease. High and low points and values at these points.
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.