1. Linear fractional function and its graph
A function of the form y = P(x) / Q(x), where P(x) and Q(x) are polynomials, is called a fractional rational function.
You are probably already familiar with the concept of rational numbers. Similarly rational functions are functions that can be represented as a quotient of two polynomials.
If a fractional rational function is a quotient of two linear functions - polynomials of the first degree, i.e. view function
y = (ax + b) / (cx + d), then it is called fractional linear.
Note that in the function y = (ax + b) / (cx + d), c ≠ 0 (otherwise the function becomes linear y = ax/d + b/d) and that a/c ≠ b/d (otherwise the function is a constant ). The linear-fractional function is defined for all real numbers, except for x = -d/c. Graphs of linear-fractional functions do not differ in form from the graph you know y = 1/x. The curve that is the graph of the function y = 1/x is called hyperbole. With an unlimited increase in x in absolute value, the function y = 1/x decreases indefinitely in absolute value and both branches of the graph approach the abscissa axis: the right one approaches from above, and the left one approaches from below. The lines approached by the branches of a hyperbola are called its asymptotes.
Example 1
y = (2x + 1) / (x - 3).
Solution.
Let's select the integer part: (2x + 1) / (x - 3) = 2 + 7 / (x - 3).
Now it is easy to see that the graph of this function is obtained from the graph of the function y = 1/x by the following transformations: shift by 3 unit segments to the right, stretch along the Oy axis by 7 times and shift by 2 unit segments up.
Any fraction y = (ax + b) / (cx + d) can be written in the same way, highlighting the “whole part”. Consequently, the graphs of all linear-fractional functions are hyperbolas shifted along the coordinate axes in various ways and stretched along the Oy axis.
To build a graph of some arbitrary linear fractional function it is not at all necessary to transform the fraction that defines this function. Since we know that the graph is a hyperbola, it will be enough to find the lines to which its branches approach - the hyperbola asymptotes x = -d/c and y = a/c.
Example 2
Find the asymptotes of the graph of the function y = (3x + 5)/(2x + 2).
Solution.
The function is not defined, for x = -1. Hence, the line x = -1 serves as a vertical asymptote. To find the horizontal asymptote, let's find out what the values of the function y(x) approach when the argument x increases in absolute value.
To do this, we divide the numerator and denominator of the fraction by x:
y = (3 + 5/x) / (2 + 2/x).
As x → ∞ the fraction tends to 3/2. Hence, the horizontal asymptote is the straight line y = 3/2.
Example 3
Plot the function y = (2x + 1)/(x + 1).
Solution.
We select the “whole part” of the fraction:
(2x + 1) / (x + 1) = (2x + 2 - 1) / (x + 1) = 2(x + 1) / (x + 1) - 1/(x + 1) =
2 – 1/(x + 1).
Now it is easy to see that the graph of this function is obtained from the graph of the function y = 1/x by the following transformations: a shift of 1 unit to the left, a symmetric display with respect to Ox, and a shift of 2 unit intervals up along the Oy axis.
Domain of definition D(y) = (-∞; -1)ᴗ(-1; +∞).
Range of values E(y) = (-∞; 2)ᴗ(2; +∞).
Intersection points with axes: c Oy: (0; 1); c Ox: (-1/2; 0). The function increases on each of the intervals of the domain of definition.
Answer: figure 1.
2. Fractional-rational function
Consider a fractional rational function of the form y = P(x) / Q(x), where P(x) and Q(x) are polynomials of degree higher than the first.
Examples of such rational functions:
y \u003d (x 3 - 5x + 6) / (x 7 - 6) or y \u003d (x - 2) 2 (x + 1) / (x 2 + 3).
If the function y = P(x) / Q(x) is a quotient of two polynomials of degree higher than the first, then its graph will, as a rule, be more complicated, and it can sometimes be difficult to build it exactly, with all the details. However, it is often enough to apply techniques similar to those with which we have already met above.
Let the fraction be proper (n< m). Известно, что любую несократимую rational fraction can be represented, and moreover in a unique way, as a sum of a finite number of elementary fractions, the form of which is determined by expanding the denominator of the fraction Q(x) into a product of real factors:
P(x) / Q(x) \u003d A 1 / (x - K 1) m1 + A 2 / (x - K 1) m1-1 + ... + A m1 / (x - K 1) + ... +
L 1 /(x – K s) ms + L 2 /(x – K s) ms-1 + … + L ms /(x – K s) + …+
+ (B 1 x + C 1) / (x 2 +p 1 x + q 1) m1 + … + (B m1 x + C m1) / (x 2 +p 1 x + q 1) + …+
+ (M 1 x + N 1) / (x 2 + p t x + q t) m1 + ... + (M m1 x + N m1) / (x 2 + p t x + q t).
Obviously, the graph of a fractional rational function can be obtained as the sum of graphs of elementary fractions.
Plotting fractional rational functions
Consider several ways to plot a fractional-rational function.
Example 4
Plot the function y = 1/x 2 .
Solution.
We use the graph of the function y \u003d x 2 to plot the graph y \u003d 1 / x 2 and use the method of "dividing" the graphs.
Domain D(y) = (-∞; 0)ᴗ(0; +∞).
Range of values E(y) = (0; +∞).
There are no points of intersection with the axes. The function is even. Increases for all x from the interval (-∞; 0), decreases for x from 0 to +∞.
Answer: figure 2.
Example 5
Plot the function y = (x 2 - 4x + 3) / (9 - 3x).
Solution.
Domain D(y) = (-∞; 3)ᴗ(3; +∞).
y \u003d (x 2 - 4x + 3) / (9 - 3x) \u003d (x - 3) (x - 1) / (-3 (x - 3)) \u003d - (x - 1) / 3 \u003d -x / 3 + 1/3.
Here we used the technique of factorization, reduction and reduction to a linear function.
Answer: figure 3.
Example 6
Plot the function y \u003d (x 2 - 1) / (x 2 + 1).
Solution.
The domain of definition is D(y) = R. Since the function is even, the graph is symmetrical about the y-axis. Before plotting, we again transform the expression by highlighting the integer part:
y \u003d (x 2 - 1) / (x 2 + 1) \u003d 1 - 2 / (x 2 + 1).
Note that the selection of the integer part in the formula of a fractional-rational function is one of the main ones when plotting graphs.
If x → ±∞, then y → 1, i.e., the line y = 1 is a horizontal asymptote.
Answer: figure 4.
Example 7
Consider the function y = x/(x 2 + 1) and try to find exactly its largest value, i.e. most high point right half of the graph. To accurately build this graph, today's knowledge is not enough. It is obvious that our curve cannot "climb" very high, since the denominator quickly begins to “overtake” the numerator. Let's see if the value of the function can be equal to 1. To do this, you need to solve the equation x 2 + 1 \u003d x, x 2 - x + 1 \u003d 0. This equation has no real roots. So our assumption is wrong. To find the most great importance function, you need to find out for which largest A the equation A \u003d x / (x 2 + 1) will have a solution. Let's replace the original equation with a quadratic one: Ax 2 - x + A = 0. This equation has a solution when 1 - 4A 2 ≥ 0. From here we find highest value A = 1/2. 
Answer: Figure 5, max y(x) = ½.
Do you have any questions? Don't know how to build function graphs?
To get the help of a tutor - register.
The first lesson is free!
site, with full or partial copying of the material, a link to the source is required.
A function graph is a visual representation of the behavior of some function on 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
- 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.
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))).
Drawing points on the coordinate plane
- -1: -1 + 2 = 1
- 0: 0 +2 = 2
- 1: 1 + 2 = 3
-
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).
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.
Plotting a complex function
-
Find the coordinates of several points and plot them on the coordinate plane. Simply select multiple x values and plug them into the function to find the corresponding y values. Then plot the points on the coordinate plane. The more complex the function, the more points you need to find and apply. In most cases, substitute x = -1; x = 0; x = 1, but if the function is complex, find three points on each side of the origin.
- In case of function y = 5 x 2 + 6 (\displaystyle y=5x^(2)+6) substitute the following "x" values: -1, 0, 1, -2, 2, -10, 10. You will get enough points.
- Choose your x values wisely. In our example, it is easy to understand that the negative sign does not play a role: the value of "y" at x \u003d 10 and at x \u003d -10 will be the same.
-
- If you don't know what to do, start by substituting into a function different meanings"x" to find the "y" values (and thus the coordinates of the points). Theoretically, a function graph can be constructed using only this method (if, of course, we substitute an infinite variety of x values).
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 fractional expression. Therefore, it is recommended to use common sense:
Your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please read our privacy policy and let us know if you have any questions.
Collection and use of personal information
Personal information refers to data that can be used to identify or contact a specific person.
You may be asked to provide your personal information at any time when you contact us.
The following are some examples of the types of personal information we may collect and how we may use such information.
What personal information we collect:
- When you submit an application on the site, we may collect various information, including your name, phone number, address Email etc.
How we use your personal information:
- Collected by us personal information allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
- From time to time, we may use your personal information to send you important notices and communications.
- We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
- If you enter a prize draw, contest or similar incentive, we may use the information you provide to administer such programs.
Disclosure to third parties
We do not disclose information received from you to third parties.
Exceptions:
- In the event that it is necessary - in accordance with the law, judicial order, in legal proceedings, and / or based on public requests or requests from state bodies in the territory of the Russian Federation - disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public interest reasons.
- In the event of a reorganization, merger or sale, we may transfer the personal information we collect to the relevant third party successor.
Protection of personal information
We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as from unauthorized access, disclosure, alteration and destruction.
Maintaining your privacy at the company level
To ensure that your personal information is secure, we communicate privacy and security practices to our employees and strictly enforce privacy practices.
The length of the segment on the coordinate axis is found by the formula:
The length of the segment on the coordinate plane is sought by the formula:
To find the length of a segment in a three-dimensional coordinate system, the following formula is used:
The coordinates of the middle of the segment (for the coordinate axis only the first formula is used, for the coordinate plane - the first two formulas, for the three-dimensional coordinate system - all three formulas) are calculated by the formulas:
Function is a correspondence of the form y= f(x) between variables, due to which each considered value of some variable x(argument or independent variable) corresponds to a certain value of another variable, y(dependent variable, sometimes this value is simply called the value of the function). Note that the function assumes that one value of the argument X there can only be one value of the dependent variable at. However, the same value at can be obtained with various X.
Function scope are all values of the independent variable (function argument, usually X) for which the function is defined, i.e. its meaning exists. The domain of definition is indicated D(y). By and large, you are already familiar with this concept. The scope of a function is otherwise called the domain of valid values, or ODZ, which you have been able to find for a long time.
Function range are all possible values of the dependent variable of this function. Denoted E(at).
Function rises on the interval on which the larger value of the argument corresponds to the larger value of the function. Function Decreasing on the interval on which the larger value of the argument corresponds to the smaller value of the function.
Function intervals are the intervals of the independent variable at which the dependent variable retains its positive or negative sign.
Function zeros are those values of the argument for which the value of the function is equal to zero. At these points, the graph of the function intersects the abscissa axis (OX axis). Very often, the need to find the zeros of a function means simply solving the equation. Also, often the need to find intervals of constant sign means the need to simply solve the inequality.
Function y = f(x) are called even X
This means that for any opposite values of the argument, the values of the even function are equal. Schedule even function always symmetrical about the y-axis of the y.
Function y = f(x) are called odd, if it is defined on a symmetric set and for any X from the domain of definition the equality is fulfilled:
This means that for any opposite values of the argument, the values of the odd function are also opposite. The graph of an odd function is always symmetrical about the origin.
The sum of the roots of an even and odd features(points of intersection of the x-axis OX) is always zero, because for every positive root X has a negative root X.
It is important to note that some function does not have to be even or odd. There are many functions that are neither even nor odd. Such functions are called functions general view , and none of the above equalities or properties hold for them.
Linear function is called a function that can be given by the formula:
The graph of a linear function is a straight line and in the general case looks like this (an example is given for the case when k> 0, in this case the function is increasing; for the occasion k < 0 функция будет убывающей, т.е. прямая будет наклонена в другую сторону - слева направо):
Graph of Quadratic Function (Parabola)
The graph of a parabola is given by a quadratic function:
A quadratic function, like any other function, intersects the OX axis at the points that are its roots: ( x 1 ; 0) and ( x 2; 0). If there are no roots, then the quadratic function does not intersect the OX axis, if there is one root, then at this point ( x 0; 0) the quadratic function only touches the OX axis, but does not intersect it. A quadratic function always intersects the OY axis at a point with coordinates: (0; c). The graph of a quadratic function (parabola) may look like this (the figure shows examples that far from exhaust all possible types of parabolas):
Wherein:
- if the coefficient a> 0, in the function y = ax 2 + bx + c, then the branches of the parabola are directed upwards;
- if a < 0, то ветви параболы направлены вниз.
Parabola vertex coordinates can be calculated using the following formulas. X tops (p- in the figures above) of a parabola (or the point at which the square trinomial reaches its maximum or minimum value):
Y tops (q- in the figures above) of a parabola or the maximum if the branches of the parabola are directed downwards ( a < 0), либо минимальное, если ветви параболы направлены вверх (a> 0), value square trinomial:
Graphs of other functions
power function
Here are some examples of graphs of power functions:
Inversely proportional dependence call the function given by the formula:
Depending on the sign of the number k An inversely proportional graph can have two fundamental options:
Asymptote is the line to which the line of the graph of the function approaches infinitely close, but does not intersect. Asymptotes for Graphs inverse proportionality shown in the figure above are the coordinate axes to which the graph of the function approaches infinitely close, but does not intersect them.
exponential function with base A call the function given by the formula:
a schedule exponential function can have two fundamental options (we will also give examples, see below):
logarithmic function call the function given by the formula:
Depending on whether the number is greater or less than one a The graph of a logarithmic function can have two fundamental options:
Function Graph y = |x| as follows:
Graphs of periodic (trigonometric) functions
Function at = f(x) is called periodical, if there exists such a non-zero number T, What f(x + T) = f(x), for anyone X out of function scope f(x). If the function f(x) is periodic with period T, then the function:
Where: A, k, b are constant numbers, and k not equal to zero, also periodic with a period T 1 , which is determined by the formula:
Most examples of periodic functions are trigonometric functions. Here are the graphs of the main trigonometric functions. The following figure shows part of the graph of the function y= sin x(the whole graph continues indefinitely to the left and right), the graph of the function y= sin x called sinusoid:
Function Graph y= cos x called cosine wave. This graph is shown in the following figure. Since the graph of the sine, it continues indefinitely along the OX axis to the left and to the right:
Function Graph y=tg x called tangentoid. This graph is shown in the following figure. Like the graphs of other periodic functions, this graph repeats indefinitely along the OX axis to the left and to the right.
And finally, the graph of the function y=ctg x called cotangentoid. This graph is shown in the following figure. Like the graphs of other periodic and trigonometric functions, this graph repeats indefinitely along the OX axis to the left and to the right.
Successful, diligent and responsible implementation of these three points will allow you to show an excellent result on the CT, the maximum of what you are capable of.
Found an error?
If you think you have found an error in training materials, then write, please, about it by mail. You can also report a bug in social network(). In the letter, indicate the subject (physics or mathematics), the name or number of the topic or test, the number of the task, or the place in the text (page) where, in your opinion, there is an error. Also describe what the alleged error is. Your letter will not go unnoticed, the error will either be corrected, or you will be explained why it is not a mistake.
