Graph of linear function y kx. How to solve linear functions
1) Function domain and function range.
The domain of a function is the set of all valid valid argument values x(variable x), for which the function y = f(x) determined. The range of a function is the set of all real values y, which the function accepts.
In elementary mathematics, functions are studied only on the set of real numbers.
2) Function zeros.
Function zero is the value of the argument at which the value of the function is equal to zero.
3) Intervals of constant sign of a function.
Intervals of constant sign of a function are sets of argument values on which the function values are only positive or only negative.
4) Monotonicity of the function.
An increasing function (in a certain interval) is a function for which higher value the argument from this interval corresponds to a larger value of the function.
A decreasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.
5) Even (odd) function.
An even function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality f(-x) = f(x). Schedule even function symmetrical about the ordinate axis.
An odd function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality is true f(-x) = - f(x). Schedule odd function symmetrical about the origin.
6) Limited and unlimited functions.
A function is called bounded if there is such a positive number M such that |f(x)| ≤ M for all values of x. If such a number does not exist, then the function is unlimited.
7) Periodicity of the function.
A function f(x) is periodic if there is a non-zero number T such that for any x from the domain of definition of the function the following holds: f(x+T) = f(x). This smallest number is called the period of the function. All trigonometric functions are periodic. (Trigonometric formulas).
19. Basic elementary functions, their properties and graphs. Application of functions in economics.
Basic elementary functions. Their properties and graphs
1. Linear function.
Linear function is called a function of the form , where x is a variable, a and b are real numbers.
Number A called the slope of the line, it is equal to the tangent of the angle of inclination of this line to the positive direction of the x-axis. The graph of a linear function is a straight line. It is defined by two points.
Properties of a Linear Function
1. Domain of definition - the set of all real numbers: D(y)=R
2. The set of values is the set of all real numbers: E(y)=R
3. The function takes a zero value when or.
4. The function increases (decreases) over the entire domain of definition.
5. Linear function continuous over the entire domain of definition, differentiable and .
2. Quadratic function.
A function of the form, where x is a variable, coefficients a, b, c are real numbers, is called quadratic
Instructions
If the graph is a straight line passing through the origin of coordinates and forming an angle α with the OX axis (the angle of inclination of the straight line to the positive semi-axis OX). The function describing this line will have the form y = kx. The proportionality coefficient k is equal to tan α. If a straight line passes through the 2nd and 4th coordinate quarters, then k< 0, и является убывающей, если через 1-ю и 3-ю, то k >0 and the function is increasing. Let it be a straight line located in various ways relative to the coordinate axes. This is a linear function and has the form y = kx + b, where the variables x and y are to the first power, and k and b can be either positive or negative. negative values or equal to zero. The line is parallel to the line y = kx and cuts off at the axis |b| units. If the line is parallel to the abscissa axis, then k = 0, if the ordinate axis, then the equation has the form x = const.
A curve consisting of two branches located in different quarters and symmetrical relative to the origin of coordinates is a hyperbola. This graph is the inverse dependence of the variable y on x and is described by the equation y = k/x. Here k ≠ 0 is the proportionality coefficient. Moreover, if k > 0, the function decreases; if k< 0 - функция возрастает. Таким образом, областью определения функции является вся числовая прямая, кроме x = 0. Ветви приближаются к осям координат как к своим асимптотам. С уменьшением |k| ветки гиперболы все больше «вдавливаются» в координатные углы.
The quadratic function has the form y = ax2 + bx + c, where a, b and c are constant quantities and a 0. If the condition b = c = 0 is met, the function equation looks like y = ax2 ( simplest case), and its graph is a parabola passing through the origin. The graph of the function y = ax2 + bx + c has the same shape as the simplest case of the function, but its vertex (the point of intersection with the OY axis) does not lie at the origin.
A parabola is also the graph of a power function expressed by the equation y = xⁿ, if n is any even number. If n is any odd number, the graph of such a power function will look like a cubic parabola.
If n is any , the function equation takes the form. The graph of the function for odd n will be a hyperbola, and for even n their branches will be symmetrical with respect to the op axis.
Also in school years The functions are studied in detail and their graphs are constructed. But, unfortunately, they practically do not teach how to read the graph of a function and find its type from the presented drawing. It's actually quite simple if you remember the basic types of functions.
Instructions
If the presented graph is , which is through the origin of coordinates and with the OX axis the angle α (which is the angle of inclination of the straight line to the positive semi-axis), then the function describing such a straight line will be presented as y = kx. In this case, the proportionality coefficient k is equal to the tangent of the angle α.
If a given line passes through the second and fourth coordinate quarters, then k is equal to 0 and the function increases. Let the presented graph be a straight line located in any way relative to the coordinate axes. Then the function of such graphic arts will be linear, which is represented by the form y = kx + b, where the variables y and x are in the first, and b and k can take both negative and positive values or .
If the line is parallel to the line with the graph y = kx and cuts off b units on the ordinate axis, then the equation has the form x = const, if the graph is parallel to the abscissa axis, then k = 0.
A curved line that consists of two branches, symmetrical about the origin and located in different quarters, is a hyperbola. Such a graph shows the inverse dependence of the variable y on the variable x and is described by an equation of the form y = k/x, where k should not be equal to zero, since it is a coefficient of inverse proportionality. Moreover, if the value of k is greater than zero, the function decreases; if k less than zero– increases.
If the proposed graph is a parabola passing through the origin, its function, subject to the condition that b = c = 0, will have the form y = ax2. This is the simplest case of a quadratic function. The graph of a function of the form y = ax2 + bx + c will have the same form as the simplest case, however, the vertex (the point where the graph intersects the ordinate axis) will not be at the origin. In a quadratic function, represented by the form y = ax2 + bx + c, the values of a, b and c are constant, while a is not equal to zero.
A parabola can also be the graph of a power function expressed by an equation of the form y = xⁿ only if n is any even number. If the value of n is an odd number, such a graph of a power function will be represented by a cubic parabola. In case the variable n is any negative number, the equation of the function takes the form .
Video on the topic
The coordinate of absolutely any point on the plane is determined by its two quantities: along the abscissa axis and the ordinate axis. The collection of many such points represents the graph of the function. From it you can see how the Y value changes depending on the change in the X value. You can also determine in which section (interval) the function increases and in which it decreases.
Instructions
What can you say about a function if its graph is a straight line? See if this line passes through the coordinate origin point (that is, the one where the X and Y values are equal to 0). If it passes, then such a function is described by the equation y = kx. It is easy to understand that the larger the value of k, the closer to the ordinate axis this straight line will be located. And the Y axis itself actually corresponds infinitely of great importance k.
A linear function is a function of the form
x-argument (independent variable),
y-function (dependent variable),
k and b are some constant numbers
The graph of a linear function is straight.
To create a graph it is enough two points, because through two points you can draw a straight line and, moreover, only one.
If k˃0, then the graph is located in the 1st and 3rd coordinate quarters. If k˂0, then the graph is located in the 2nd and 4th coordinate quarters.
The number k is called the slope of the straight graph of the function y(x)=kx+b. If k˃0, then the angle of inclination of the straight line y(x)= kx+b to the positive direction Ox is acute; if k˂0, then this angle is obtuse.
Coefficient b shows the point of intersection of the graph with the op-amp axis (0; b).
y(x)=k∙x-- special case A typical function is called direct proportionality. The graph is a straight line passing through the origin, so one point is enough to construct this graph.
Graph of a Linear Function
Where coefficient k = 3, therefore
The graph of the function will increase and have an acute angle with the Ox axis because coefficient k has a plus sign.
OOF linear function
OPF of a linear function
Except in the case where
Also a linear function of the form
Is a function general view.
B) If k=0; b≠0,
In this case, the graph is a straight line parallel to the Ox axis and passing through the point (0; b).
B) If k≠0; b≠0, then the linear function has the form y(x)=k∙x+b.
Example 1 . Graph the function y(x)= -2x+5
Example 2 . Let's find the zeros of the function y=3x+1, y=0;
– zeros of the function.
Answer: or (;0)
Example 3 . Determine the value of the function y=-x+3 for x=1 and x=-1
y(-1)=-(-1)+3=1+3=4
Answer: y_1=2; y_2=4.
Example 4 . Determine the coordinates of their intersection point or prove that the graphs do not intersect. Let the functions y 1 =10∙x-8 and y 2 =-3∙x+5 be given.
If the graphs of functions intersect, then the values of the functions at this point are equal
Substitute x=1, then y 1 (1)=10∙1-8=2.
Comment. You can also substitute the resulting value of the argument into the function y 2 =-3∙x+5, then we get the same answer y 2 (1)=-3∙1+5=2.
y=2- ordinate of the intersection point.
(1;2) - the point of intersection of the graphs of the functions y=10x-8 and y=-3x+5.
Answer: (1;2)
Example 5 .
Construct graphs of the functions y 1 (x)= x+3 and y 2 (x)= x-1.
You can notice that the coefficient k=1 for both functions.
From the above it follows that if the coefficients of a linear function are equal, then their graphs in the coordinate system are located parallel.
Example 6 .
Let's build two graphs of the function.
The first graph has the formula
The second graph has the formula
IN in this case Before us is a graph of two lines intersecting at the point (0;4). This means that the coefficient b, which is responsible for the height of the rise of the graph above the Ox axis, if x = 0. This means we can assume that the b coefficient of both graphs is equal to 4.
Editors: Ageeva Lyubov Aleksandrovna, Gavrilina Anna Viktorovna
Instructions
There are several ways to solve linear functions. Let's list the most of them. Most often used step by step method substitutions. In one of the equations it is necessary to express one variable in terms of another and substitute it into another equation. And so on until only one variable remains in one of the equations. To solve it, you need to leave a variable on one side of the equal sign (it can be with a coefficient), and on the other side of the equal sign all the numerical data, not forgetting to change the sign of the number to the opposite one when transferring. Having calculated one variable, substitute it into other expressions and continue calculations using the same algorithm.
For example, let's take a linear system functions, consisting of two equations:
2x+y-7=0;
x-y-2=0.
It is convenient to express x from the second equation:
x=y+2.
As you can see, when transferring from one part of the equality to another, the sign of y and variables changed, as was described above.
We substitute the resulting expression into the first equation, thus excluding the variable x from it:
2*(y+2)+y-7=0.
Expanding the brackets:
2y+4+y-7=0.
We put together variables and numbers and add them up:
3у-3=0.
We move it to the right side of the equation and change the sign:
3y=3.
Divide by the total coefficient, we get:
y=1.
We substitute the resulting value into the first expression:
x=y+2.
We get x=3.
Another way to solve similar ones is to add two equations term by term to get a new one with one variable. The equation can be multiplied by a certain coefficient, the main thing is to multiply each member of the equation and not forget, and then add or subtract one equation from. This method is very economical when finding a linear functions.
Let’s take the already familiar system of equations with two variables:
2x+y-7=0;
x-y-2=0.
It is easy to notice that the coefficient of the variable y is identical in the first and second equations and differs only in sign. This means that when we add these two equations term by term, we get a new one, but with one variable.
2x+x+y-y-7-2=0;
3x-9=0.
We transfer the numerical data to the right side of the equation, changing the sign:
3x=9.
We find a common factor equal to the coefficient at x and divide both sides of the equation by it:
x=3.
The result can be substituted into any of the system equations to calculate y:
x-y-2=0;
3-у-2=0;
-y+1=0;
-y=-1;
y=1.
You can also calculate data by creating an accurate graph. To do this you need to find zeros functions. If one of the variables is equal to zero, then such a function is called homogeneous. Having solved such equations, you will get two points necessary and sufficient to construct a straight line - one of them will be located on the x-axis, the other on the y-axis.
We take any equation of the system and substitute the value x=0 there:
2*0+y-7=0;
We get y=7. Thus, the first point, let's call it A, will have coordinates A(0;7).
In order to calculate a point lying on the x-axis, it is convenient to substitute the value y=0 into the second equation of the system:
x-0-2=0;
x=2.
The second point (B) will have coordinates B (2;0).
We mark the obtained points on the coordinate grid and draw a straight line through them. If you plot it fairly accurately, other values of x and y can be calculated directly from it.