1. Fractional linear 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. Likewise rational functions are functions that can be represented as the quotient of two polynomials.
If a fractional rational function is the quotient of two linear functions - polynomials of the first degree, i.e. function of the form
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 constant ). The fractional linear function is defined for all real numbers, except x = -d/c. Graphs of fractional linear functions do not differ in shape from the graph y = 1/x you know. A curve that is a 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 unlimited in absolute value and both branches of the graph approach the abscissa: the right one approaches from above, and the left one from below. The lines to which the branches of a hyperbola approach are called its asymptotes.
Example 1.
y = (2x + 1) / (x – 3).
Solution.
Let's select the whole 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, stretching along the Oy axis 7 times and shifting by 2 unit segments upward.
Any fraction y = (ax + b) / (cx + d) can be written in a similar way, highlighting the “integer part”. Consequently, the graphs of all fractional linear functions are hyperbolas, shifted in various ways along the coordinate axes and stretched along the Oy axis.
To construct a graph of any arbitrary fractional linear function It is not at all necessary to transform the fraction defining this function. Since we know that the graph is a hyperbola, it will be enough to find the straight lines to which its branches approach - the asymptotes of the hyperbola 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, at x = -1. This means that the straight 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, divide the numerator and denominator of the fraction by x:
y = (3 + 5/x) / (2 + 2/x).
As x → ∞ the fraction will tend to 3/2. Means, horizontal asymptote– this is the straight line y = 3/2.
Example 3.
Graph the function y = (2x + 1)/(x + 1).
Solution.
Let’s 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 by 1 unit to the left, a symmetrical display with respect to Ox and a shift by 2 unit segments up along the Oy axis.
Domain 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 at each interval 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 first.
Examples of such rational functions:
y = (x 3 – 5x + 6) / (x 7 – 6) or y = (x – 2) 2 (x + 1) / (x 2 + 3).
If the function y = P(x) / Q(x) represents the quotient of two polynomials of degree higher than the first, then its graph will, as a rule, be more complex, and it can sometimes be difficult to construct it accurately, with all the details. However, it is often enough to use techniques similar to those we have already introduced above.
Let the fraction be a proper fraction (n< m). Известно, что любую несократимую рациональную дробь можно представить, и притом единственным образом, в виде суммы конечного числа элементарных дробей, вид которых определяется разложением знаменателя дроби Q(x) в произведение действительных сомножителей:
P(x)/Q(x) = 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 graphs of fractional rational functions
Let's consider several ways to construct graphs of a fractional rational function.
Example 4.
Graph the function y = 1/x 2 .
Solution.
We use the graph of the function y = x 2 to construct a graph of y = 1/x 2 and use the technique 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.
Graph the function y = (x 2 – 4x + 3) / (9 – 3x).
Solution.
Domain D(y) = (-∞; 3)ᴗ(3; +∞).
y = (x 2 – 4x + 3) / (9 – 3x) = (x – 3)(x – 1) / (-3(x – 3)) = -(x – 1)/3 = -x/ 3 + 1/3.
Here we used the technique of factorization, reduction and reduction to a linear function.
Answer: Figure 3.
Example 6.
Graph the function y = (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 ordinate. Before building a graph, let’s transform the expression again, highlighting the whole part:
y = (x 2 – 1)/(x 2 + 1) = 1 – 2/(x 2 + 1).
Note that isolating the integer part in the formula of a fractional rational function is one of the main ones when constructing graphs.
If x → ±∞, then y → 1, i.e. the straight line y = 1 is a horizontal asymptote.
Answer: Figure 4.
Example 7.
Let's consider the function y = x/(x 2 + 1) and try to accurately find its largest value, i.e. the highest point on the right half of the graph. To accurately construct this graph, today's knowledge is not enough. Obviously, our curve cannot “rise” very high, because 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, we need to solve the equation x 2 + 1 = x, x 2 – x + 1 = 0. This equation has no real roots. This means our assumption is incorrect. To find the most great value function, you need to find out at what largest A the equation A = x/(x 2 + 1) will have a solution. Let's replace the original equation with a quadratic one: Аx 2 – x + А = 0. This equation has a solution when 1 – 4А 2 ≥ 0. From here we find highest value A = 1/2. 
Answer: Figure 5, max y(x) = ½.
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A function graph is a visual representation of the behavior of a function on a 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 certain 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
- If the slope is negative, the function is decreasing.
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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
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Define a function. The function is denoted as f(x). All possible values the variable "y" is 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: to the right (from 0) are plotted positive numbers, and on the left are negative. 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
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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 fractional expression. Therefore, it is recommended to use common sense:
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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))).
The exponent is denoted as , or .
Number e
The basis of the exponent degree is number e. This irrational number. It is approximately equal
e ≈ 2,718281828459045...
The number e is determined through the limit of the sequence. This is the so-called second wonderful limit:
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The number e can also be represented as a series:
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Exponential graph
Exponential graph, y = e x .The graph shows the exponential e to a degree X.
y (x) = e x
The graph shows that the exponent increases monotonically.
Formulas
The basic formulas are the same as for exponential function with power base e.
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Expression of an exponential function with an arbitrary base of degree a through an exponential:
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Private values
Let y (x) = e x. Then
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Exponent Properties
The exponent has the properties of an exponential function with a power base e > 1 .
Domain, set of values
Exponent y (x) = e x defined for all x.
Its domain of definition:
- ∞ < x + ∞
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Its many meanings:
0
< y < + ∞
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Extremes, increasing, decreasing
The exponential is a monotonically increasing function, so it has no extrema. Its main properties are presented in the table.
Inverse function
The inverse of the exponent is the natural logarithm.
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Derivative of the exponent
Derivative e to a degree X equal to e to a degree X
:
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Derivative of nth order:
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Deriving formulas > > >
Integral
Complex numbers
Actions with complex numbers carried out using Euler's formulas:
,
where is the imaginary unit:
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Expressions through hyperbolic functions
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Expressions using trigonometric functions
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Power series expansion
Used literature:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
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