>>Mathematics: Linear function and its graph
Linear function and its graph
The algorithm for constructing a graph of the equation ax + by + c = 0, which we formulated in § 28, for all its clarity and certainty, mathematicians do not really like. They usually make claims about the first two steps of the algorithm. Why, they say, solve the equation twice for the variable y: first ax1 + by + c = O, then ax1 + by + c = O? Isn’t it better to immediately express y from the equation ax + by + c = 0, then it will be easier to carry out calculations (and, most importantly, faster)? Let's check. Let's consider first the equation 3x - 2y + 6 = 0 (see example 2 from § 28).
Giving x specific values, it is easy to calculate the corresponding values of y. For example, when x = 0 we get y = 3; at x = -2 we have y = 0; for x = 2 we have y = 6; for x = 4 we get: y = 9.
You see how easily and quickly the points (0; 3), (- 2; 0), (2; 6) and (4; 9) were found, which were highlighted in example 2 from § 28.
In the same way, the equation bx - 2y = 0 (see example 4 from § 28) could be transformed to the form 2y = 16 -3x. further y = 2.5x; it is not difficult to find points (0; 0) and (2; 5) satisfying this equation.
Finally, the equation 3x + 2y - 16 = 0 from the same example can be transformed to the form 2y = 16 -3x and then it is not difficult to find points (0; 0) and (2; 5) that satisfy it.
Let us now consider these transformations in general form.
Thus, linear equation (1) with two variables x and y can always be transformed to the form
y = kx + m,(2) where k,m are numbers (coefficients), and .
This private view linear equation will be called a linear function.
Using equality (2), it is easy to specify a specific x value and calculate the corresponding y value. Let, for example,
y = 2x + 3. Then:
if x = 0, then y = 3;
if x = 1, then y = 5;
if x = -1, then y = 1;
if x = 3, then y = 9, etc.
Typically these results are presented in the form tables:
The values of y from the second row of the table are called the values of the linear function y = 2x + 3, respectively, at the points x = 0, x = 1, x = -1, x = -3.
In equation (1) the variables hnu are equal, but in equation (2) they are not: we assign specific values to one of them - variable x, while the value of variable y depends on the selected value of variable x. Therefore, we usually say that x is the independent variable (or argument), y is the dependent variable.
Note: linear function- This special type linear equation with two variables. Equation graph y - kx + m, like any linear equation with two variables, is a straight line - it is also called the graph of the linear function y = kx + m. Thus, the following theorem is valid.
Example 1. Construct a graph of the linear function y = 2x + 3.
Solution. Let's make a table:
In the second situation, the independent variable x, which, as in the first situation, denotes the number of days, can only take the values 1, 2, 3, ..., 16. Indeed, if x = 16, then using the formula y = 500 - 30x we find : y = 500 - 30 16 = 20. This means that already on the 17th day it will not be possible to remove 30 tons of coal from the warehouse, since by this day only 20 tons will remain in the warehouse and the process of coal removal will have to be stopped. Therefore, the refined mathematical model of the second situation looks like this:
y = 500 - ZOD:, where x = 1, 2, 3, .... 16.
In the third situation, independent variable x can theoretically take on any non-negative value (for example, x value = 0, x value = 2, x value = 3.5, etc.), but in practice a tourist cannot walk with constant speed without sleep or rest for as long as desired. So we needed to make reasonable restrictions on x, say 0< х < 6 (т. е. турист идет не более 6 ч).
Recall that the geometric model of the non-strict double inequality 0< х < 6 служит отрезок (рис. 37). Значит, уточненная модель третьей ситуации выглядит так: у = 15 + 4х, где х принадлежит отрезку .
Let us agree to write instead of the phrase “x belongs to the set X” (read: “element x belongs to the set X”, e is the sign of membership). As you can see, our acquaintance with mathematical language is constantly ongoing.
If the linear function y = kx + m should be considered not for all values of x, but only for values of x from a certain numerical interval X, then they write:
Example 2. Graph a linear function:
Solution, a) Let's make a table for the linear function y = 2x + 1
Let's build on the coordinate xOy plane points (-3; 7) and (2; -3) and draw a straight line through them. This is a graph of the equation y = -2x: + 1. Next, select a segment connecting the constructed points (Fig. 38). This segment is the graph of the linear function y = -2x+1, wherexe [-3, 2].
They usually say this: we have plotted a linear function y = - 2x + 1 on the segment [- 3, 2].
b) How does this example differ from the previous one? The linear function is the same (y = -2x + 1), which means that the same straight line serves as its graph. But - be careful! - this time x e (-3, 2), i.e. the values x = -3 and x = 2 are not considered, they do not belong to the interval (- 3, 2). How did we mark the ends of an interval on a coordinate line? Light circles (Fig. 39), we talked about this in § 26. Similarly, points (- 3; 7) and B; - 3) will have to be marked on the drawing with light circles. This will remind us that only those points of the line y = - 2x + 1 are taken that lie between the points marked with circles (Fig. 40). However, sometimes in such cases they use arrows rather than light circles (Fig. 41). This is not fundamental, the main thing is to understand what is being said.
Example 3. Find the largest and smallest values of a linear function on the segment.
Solution. Let's make a table for a linear function
Let's build on coordinate plane xОу points (0; 4) and (6; 7) and draw a straight line through them - a graph of the linear x function (Fig. 42).
We need to consider this linear function not as a whole, but on a segment, i.e. for x e.
The corresponding segment of the graph is highlighted in the drawing. We notice that the largest ordinate of the points belonging to the selected part is equal to 7 - this is highest value linear function on the segment. Usually the following notation is used: y max =7.
We note that the smallest ordinate of the points belonging to the part of the line highlighted in Figure 42 is equal to 4 - this is the smallest value of the linear function on the segment.
Usually the following notation is used: y name. = 4.
Example 4. Find y naib and y naim. for a linear function y = -1.5x + 3.5
a) on the segment; b) on the interval (1.5);
c) on a half-interval.
Solution. Let's make a table for the linear function y = -l.5x + 3.5:
Let's construct points (1; 2) and (5; - 4) on the xOy coordinate plane and draw a straight line through them (Fig. 43-47). Let us select on the constructed straight line the part corresponding to the x values from the segment (Fig. 43), from the interval A, 5) (Fig. 44), from the half-interval (Fig. 47).
a) Using Figure 43, it is easy to conclude that y max = 2 (the linear function reaches this value at x = 1), and y min. = - 4 (the linear function reaches this value at x = 5).
b) Using Figure 44, we conclude: this linear function has neither the largest nor the smallest values on a given interval. Why? The fact is that, unlike the previous case, both ends of the segment, in which the largest and smallest values were reached, are excluded from consideration.
c) Using Figure 45, we conclude that y max. = 2 (as in the first case), and lowest value the linear function does not (as in the second case).
d) Using Figure 46, we conclude: y max = 3.5 (the linear function reaches this value at x = 0), and y max. does not exist.
e) Using Figure 47, we conclude: y max. = -1 (the linear function reaches this value at x = 3), and y max. does not exist.
Example 5. Graph a linear function
y = 2x - 6. Use the graph to answer the following questions:
a) at what value of x will y = 0?
b) for what values of x will y > 0?
c) at what values of x will y< 0?
Solution. Let's make a table for the linear function y = 2x-6:
Through the points (0; - 6) and (3; 0) we draw a straight line - the graph of the function y = 2x - 6 (Fig. 48).
a) y = 0 at x = 3. The graph intersects the x axis at the point x = 3, this is the point with ordinate y = 0.
b) y > 0 for x > 3. In fact, if x > 3, then the straight line is located above the x axis, which means the ordinates corresponding points direct are positive.
c) at< 0 при х < 3. В самом деле если х < 3, то прямая расположена ниже оси х, значит, ординаты соответствующих точек прямой отрицательны. A
Please note that in this example we used the graph to solve:
a) equation 2x - 6 = 0 (we got x = 3);
b) inequality 2x - 6 > 0 (we got x > 3);
c) inequality 2x - 6< 0 (получили х < 3).
Comment. In Russian, the same object is often called differently, for example: “house”, “building”, “structure”, “cottage”, “mansion”, “barrack”, “shack”, “hut”. In mathematical language the situation is approximately the same. Say, the equality with two variables y = kx + m, where k, m are specific numbers, can be called a linear function, can be called linear equation with two variables x and y (or with two unknowns x and y), can be called a formula, can be called a relation connecting x and y, can finally be called a dependence between x and y. It doesn’t matter, the main thing is to understand that in all cases we're talking about O mathematical model y = kx + m
.
Consider the graph of the linear function shown in Figure 49, a. If we move along this graph from left to right, then the ordinates of the points on the graph are increasing all the time, as if we are “climbing up a hill.” In such cases, mathematicians use the term increase and say this: if k>0, then the linear function y = kx + m increases.
Consider the graph of the linear function shown in Figure 49, b. If we move along this graph from left to right, then the ordinates of the points on the graph are decreasing all the time, as if we are “going down a hill.” In such cases, mathematicians use the term decrease and say this: if k< О, то линейная функция у = kx + m убывает.
Linear function in life
Now let's summarize this topic. We have already become acquainted with such a concept as a linear function, we know its properties and learned how to build graphs. Also, you looked at special cases of a linear function and found out what it depends on mutual arrangement graphs of linear functions. But it turns out that in our Everyday life we also constantly intersect with this mathematical model.
Let us think about what real life situations are associated with such a concept as linear functions? And also, between what quantities or life situations perhaps establish a linear relationship?
Many of you probably don’t quite understand why they need to study linear functions, because it’s unlikely to be useful in later life. But here you are deeply mistaken, because we encounter functions all the time and everywhere. Because even a regular monthly rent is also a function that depends on many variables. And these variables include square footage, number of residents, tariffs, electricity use, etc.
Of course, the most common examples of functions linear dependence, which we have encountered are mathematics lessons.
You and I solved problems where we found the distances traveled by cars, trains, or pedestrians at a certain speed. These are linear functions of movement time. But these examples are applicable not only in mathematics, they are present in our everyday life.
The calorie content of dairy products depends on the fat content, and such a dependence is usually a linear function. For example, when the percentage of fat in sour cream increases, the calorie content of the product also increases.
Now let's do the calculations and find the values of k and b by solving the system of equations:
Now let's derive the dependency formula:
As a result, we obtained a linear relationship.
To know the speed of sound propagation depending on temperature, it is possible to find out by using the formula: v = 331 +0.6t, where v is the speed (in m/s), t is the temperature. If we draw a graph of this relationship, we will see that it will be linear, that is, it will represent a straight line.
And such practical uses knowledge in the application of linear functional dependence The list could take a long time. Starting from phone charges, hair length and growth, and even proverbs in literature. And this list goes on and on.
Calendar-thematic planning in mathematics, video in mathematics online, Mathematics at school download
A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions
Definition of a Linear Function
Let us introduce the definition of a linear function
Definition
A function of the form $y=kx+b$, where $k$ is nonzero, is called a linear function.
The graph of a linear function is a straight line. The number $k$ is called the slope of the line.
When $b=0$ the linear function is called a function of direct proportionality $y=kx$.
Consider Figure 1.
Rice. 1. Geometric meaning of the slope of a line
Let's consider triangle ABC. We see that $ВС=kx_0+b$. Let's find the point of intersection of the line $y=kx+b$ with the axis $Ox$:
\ \
So $AC=x_0+\frac(b)(k)$. Let's find the ratio of these sides:
\[\frac(BC)(AC)=\frac(kx_0+b)(x_0+\frac(b)(k))=\frac(k(kx_0+b))((kx)_0+b)=k \]
On the other hand, $\frac(BC)(AC)=tg\angle A$.
Thus, we can draw the following conclusion:
Conclusion
Geometric meaning coefficient $k$. Slope factor direct $k$ equal to tangent the angle of inclination of this straight line to the $Ox$ axis.
Study of the linear function $f\left(x\right)=kx+b$ and its graph
First, consider the function $f\left(x\right)=kx+b$, where $k > 0$.
- $f"\left(x\right)=(\left(kx+b\right))"=k>0$. Hence, this function increases throughout the entire domain of definition. There are no extreme points.
- $(\mathop(lim)_(x\to -\infty ) kx\ )=-\infty $, $(\mathop(lim)_(x\to +\infty ) kx\ )=+\infty $
- Graph (Fig. 2).
Rice. 2. Graphs of the function $y=kx+b$, for $k > 0$.
Now consider the function $f\left(x\right)=kx$, where $k
- The domain of definition is all numbers.
- The range of values is all numbers.
- $f\left(-x\right)=-kx+b$. The function is neither even nor odd.
- For $x=0,f\left(0\right)=b$. When $y=0.0=kx+b,\ x=-\frac(b)(k)$.
Intersection points with coordinate axes: $\left(-\frac(b)(k),0\right)$ and $\left(0,\ b\right)$
- $f"\left(x\right)=(\left(kx\right))"=k
- $f^("")\left(x\right)=k"=0$. Therefore, the function has no inflection points.
- $(\mathop(lim)_(x\to -\infty ) kx\ )=+\infty $, $(\mathop(lim)_(x\to +\infty ) kx\ )=-\infty $
- Graph (Fig. 3).
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 numerical data to right side equations, changing the sign:
3x=9.
We find common multiplier, equal to the coefficient, standing 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).
On coordinate grid We mark the obtained points 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.
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 chart inverse relationship variable y from 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.
The graph is also a parabola 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 inverse proportionality. Moreover, if the value of k is greater than zero, the function decreases; if k is less than zero, it 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 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.
Consider the function y=k/y. The graph of this function is a line, called a hyperbola in mathematics. General form hyperbolas are shown in the figure below. (The graph shows the function y equals k divided by x, for which k equals one.)
It can be seen that the graph consists of two parts. These parts are called branches of the hyperbola. It is also worth noting that each branch of the hyperbola approaches in one of the directions closer and closer to the coordinate axes. The coordinate axes in this case are called asymptotes.
In general, any straight lines to which the graph of a function infinitely approaches but does not reach them are called asymptotes. A hyperbola, like a parabola, has axes of symmetry. For the hyperbola shown in the figure above, this is the line y=x.
Now let's deal with two general cases hyperbole. The graph of the function y = k/x, for k ≠0, will be a hyperbola, the branches of which are located either in the first and third coordinate angles, for k>0, or in the second and fourth coordinate angles, for k<0.
Basic properties of the function y = k/x, for k>0
Graph of the function y = k/x, for k>0
5. y>0 at x>0; y6. The function decreases both on the interval (-∞;0) and on the interval (0;+∞).
10. The range of values of the function is two open intervals (-∞;0) and (0;+∞).
Basic properties of the function y = k/x, for k<0
Graph of the function y = k/x, at k<0
1. Point (0;0) is the center of symmetry of the hyperbola.
2. Coordinate axes - asymptotes of the hyperbola.
4. The domain of definition of the function is all x except x=0.
5. y>0 at x0.
6. The function increases both on the interval (-∞;0) and on the interval (0;+∞).
7. The function is not limited either from below or from above.
8. A function has neither a maximum nor a minimum value.
9. The function is continuous on the interval (-∞;0) and on the interval (0;+∞). Has a gap at x=0.