Linear function definition
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.
For $b=0$ the linear function is called the direct proportionality function $y=kx$.
Consider Figure 1.
Rice. 1. The geometric meaning of the slope of the straight line
Consider triangle ABC. We see that $BC=kx_0+b$. 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, the following conclusion can be drawn:
Conclusion
Geometric meaning of the coefficient $k$. The slope of the straight line $k$ is equal to the tangent of the slope of this straight line to the axis $Ox$.
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 $k > 0$.
- $f"\left(x\right)=(\left(kx+b\right))"=k>0$. Therefore, this function increases over 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 scope is all numbers.
- The scope is all numbers.
- $f\left(-x\right)=-kx+b$. The function is neither even nor odd.
- For $x=0,f\left(0\right)=b$. For $y=0,0=kx+b,\ x=-\frac(b)(k)$.
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).
>>Math: 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. Usually they put forward claims to the first two steps of the algorithm. Why, they say, solve the equation twice with respect to the variable y: first ax1 + bu + c = O, then axi + bu + c = O? Wouldn't it be 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. Consider first the equation 3x - 2y + 6 = 0 (see example 2 from § 28).
By giving x specific values, it is easy to calculate the corresponding y values. For example, for 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 can 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.
Similarly, the equation bx - 2y = 0 (see example 4 of § 28) could be converted to the form 2y = 16 -3x. then y = 2.5x; it is easy to find points (0; 0) and (2; 5) satisfying this equation.
Finally, the equation 3x + 2y - 16 = 0 from the same example can be converted to the form 2y = 16 -3x and then it is easy to find points (0; 0) and (2; 5) that satisfy it.
Let us now consider these transformations in general form.
Thus, the linear equation (1) with two variables x and y can always be converted to the form
y = kx + m,(2) where k,m are numbers (coefficients), and .
This particular form of the linear equation will be called a linear function.
Using equality (2), it is easy, by specifying a specific value of x, to calculate the corresponding value of y. 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.
Usually these results are presented in the form tables:
The y values from the second row of the table are called the values of the linear function y \u003d 2x + 3, respectively, at the points x \u003d 0, x \u003d 1, x \u003d -1, x \u003d -3.
In equation (1) the variables xnu are equal, but in equation (2) they are not: we assign specific values to one of them - the variable x, while the value of the variable y depends on the chosen value of the variable x. Therefore, it is usually said that x is the independent variable (or argument), y is the dependent variable.
Note that a linear function is a special kind of 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 a linear function y = kx + mp. Thus, the following theorem is true.
Example 1 Construct a graph of a linear function y \u003d 2x + 3.
Solution. Let's make a table:
In the second situation, the independent variable x, denoting, as in the first situation, the number of days, can only take on the values 1, 2, 3, ..., 16. Indeed, if x \u003d 16, then using the formula y \u003d 500 - Z0x we find : y \u003d 500 - 30 16 \u003d 20. This means that already on the 17th day it will not be possible to take out 30 tons of coal from the warehouse, since only 20 tons will remain in the warehouse by that day and the process of coal export will have to be stopped. Therefore, the refined mathematical model of the second situation looks like this:
y \u003d 500 - ZOD:, where x \u003d 1, 2, 3, .... 16.
In the third situation, independent variable x can theoretically take on any non-negative value (e.g., x value = 0, x value = 2, x value = 3.5, etc.), but in practice a tourist cannot walk at a constant speed without sleeping and resting for as long as he wants . So we had to make reasonable limits on x, say 0< х < 6 (т. е. турист идет не более 6 ч).
Recall that the geometric model of the nonstrict double inequality 0< х < 6 служит отрезок (рис. 37). Значит, уточненная модель третьей ситуации выглядит так: у = 15 + 4х, где х принадлежит отрезку .
Instead of the phrase “x belongs to the set X”, we agree to write (they read: “the element x belongs to the set X”, e is the sign of membership). As you can see, our familiarity with the mathematical language is constantly ongoing.
If the linear function y \u003d kx + m should be considered not for all values of x, but only for values of x from some numerical interval X, then they write:
Example 2. Graph a linear function:
Solution, a) Make a table for the linear function y = 2x + 1
Let's build points (-3; 7) and (2; -3) on the xOy coordinate plane and draw a straight line through them. This is the graph of the equation y \u003d -2x: + 1. Next, select the segment connecting the constructed points (Fig. 38). This segment is the graph of the linear function y \u003d -2x + 1, where xe [-3, 2].
Usually they say this: we plotted a linear function y \u003d - 2x + 1 on the segment [- 3, 2].
b) How is this example different from the previous one? The linear function is the same (y \u003d -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 the interval on the coordinate line? Light circles (Fig. 39), we talked about this in § 26. Similarly, the 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 straight line y \u003d - 2x + 1 are taken that lie between the points marked with circles (Fig. 40). However, sometimes in such cases, not light circles are used, but arrows (Fig. 41). This is not fundamental, the main thing is to understand what is at stake.
Example 3 Find the largest and smallest values of the linear function on the segment .
Solution. Let's make a table for a linear function
We construct points (0; 4) and (6; 7) on the xOy coordinate plane and draw a straight line through them - the graph of the linear x function (Fig. 42).
We need to consider this linear function not as a whole, but on the segment, i.e. for x e.
The corresponding segment of the graph is highlighted in the drawing. We note that the largest ordinate of the points belonging to the selected part is 7 - this is the largest value of the linear function on the segment . The following notation is usually used: y max = 7.
We note that the smallest ordinate of the points belonging to the part of the straight line highlighted in Figure 42 is 4 - this is the smallest value of the linear function on the segment.
Usually use the following entry: y name. = 4.
Example 4 Find y naib and y naim. for linear function y = -1.5x + 3.5
a) on the segment; b) on the interval (1.5);
c) on the half-interval .
Solution. Let's make a table for the linear function y \u003d -l, 5x + 3.5:
We construct points (1; 2) and (5; - 4) on the xOy coordinate plane and draw a straight line through them (Fig. 43-47). Let us single out on the constructed straight line the part corresponding to the values of x 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 \u003d 2 (the linear function reaches this value at x \u003d 1), and y max. = - 4 (the linear function reaches this value at x = 5).
b) Using Figure 44, we conclude that this linear function has neither the largest nor the smallest values in the 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) With the help of Figure 45 we conclude that y max. = 2 (as in the first case), while the linear function does not have the smallest value (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. Plot a Linear Function
y \u003d 2x - 6. Using the graph, answer the following questions:
a) at what value of x will y = 0?
b) for what values of x will y > 0?
c) for what values of x will y< 0?
Solution. Let's make a table for the linear function y \u003d 2x-6:
Draw a straight line through the points (0; - 6) and (3; 0) - the graph of the function y \u003d 2x - 6 (Fig. 48).
a) y \u003d 0 at x \u003d 3. The graph intersects the x axis at the point x \u003d 3, this is the point with the ordinate y \u003d 0.
b) y > 0 for x > 3. Indeed, if x > 3, then the line is located above the x-axis, which means that the ordinates of the corresponding points of the line 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
Note that in this example, we decided with the help of the graph:
a) equation 2x - 6 = 0 (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”, “hut”, “hut”. In mathematical language, the situation is about the same. Say, equality with two variables y = kx + m, where k, m are specific numbers, can be called a linear function, can be called a linear equation with two variables x and y (or with two unknowns x and y), can be called a formula, can be called can be called a relation linking x and y, one can finally call it a relationship between x and y. It does not matter, the main thing is to understand that in all cases we are talking about a mathematical model y = kx + m
.
Consider the graph of a linear function shown in Figure 49, a. If we move along this graph from left to right, then the ordinates of the graph points increase all the time, we seem to “climb up the hill”. In such cases, mathematicians use the term increase and say this: if k>0, then the linear function y \u003d kx + m increases.
Consider the graph of a linear function shown in Figure 49, b. If we move along this graph from left to right, then the ordinates of the graph points decrease all the time, we seem to be “going down the 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 real life
Now let's sum up this topic. We have already got acquainted with such a concept as a linear function, we know its properties and have learned how to build graphs. Also, you considered special cases of a linear function and learned what the relative position of the graphs of linear functions depends on. But it turns out that in our daily life we also constantly intersect with this mathematical model.
Let's think about what real life situations are associated with such a concept as linear functions? And also, between what quantities or life situations, is it possible to establish a linear relationship?
Many of you probably do not quite understand why they need to learn linear functions, because this is unlikely to be useful in later life. But here you are deeply mistaken, because we encounter functions all the time and everywhere. Since, even the usual monthly rent is also a function that depends on many variables. And these variables include the square footage, the number of residents, tariffs, electricity use, etc.
Of course, the most common examples of linear dependence functions that we have come across are math lessons.
You and I solved problems where we found the distances that cars, trains or pedestrians passed at a certain speed. These are the linear functions of the motion time. But these examples are applicable not only in mathematics, they are present in our daily life.
The calorie content of dairy products depends on fat content, and such a dependence, as a rule, is a linear function. So, for example, with an increase in the percentage of fat content in sour cream, 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 got a linear relationship.
To know the speed of sound propagation depending on temperature, it is possible to find out by applying the formula: v \u003d 331 + 0.6t, where v is the speed (in m / s), t is the temperature. If we draw a graph of this dependence, we will see that it will be linear, that is, it will represent a straight line.
And such practical uses of knowledge in the application of linear functional dependence can be listed for a long time. Starting from phone charges, hair length and height, and even proverbs in literature. And this list can be continued indefinitely.
Calendar-thematic planning in mathematics, video in mathematics online, Math at school download
A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions
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 factor:
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 a k 0
If a k>0 and b>0, then the graph of the function y=kx+b looks like:
If a k>0 and b, then the graph of the function y=kx+b looks like:
If a k, then the graph of the function y=kx+b looks like:
If a 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):
Linear function is called a function of the form y = kx + b, defined on the set of all real numbers. Here k– angular coefficient (real number), b – free member (real number), x is an independent variable.
In a particular case, if k = 0, we obtain a constant function y=b, whose graph is a straight line parallel to the Ox axis, passing through the point with coordinates (0;b).
If a b = 0, then we get the function y=kx, which is in direct proportion.
b – segment length, which cuts off the line along the Oy axis, counting from the origin.
The geometric meaning of the coefficient k – tilt angle straight to the positive direction of the Ox axis is considered to be counterclockwise.
Linear function properties:
1) The domain of a linear function is the entire real axis;
2) If a k ≠ 0, then the range of the linear function is the entire real axis. If a k = 0, then the range of the linear function consists of the number b;
3) Evenness and oddness of a linear function depend on the values of the coefficients k and b.
a) b ≠ 0, k = 0, Consequently, y = b is even;
b) b = 0, k ≠ 0, Consequently y = kx is odd;
c) b ≠ 0, k ≠ 0, Consequently y = kx + b is a general function;
d) b = 0, k = 0, Consequently y = 0 is both an even and an odd function.
4) A linear function does not have the property of periodicity;
5) Intersection points with coordinate axes:
Ox: y = kx + b = 0, x = -b/k, Consequently (-b/k; 0)- point of intersection with the abscissa axis.
Oy: y=0k+b=b, Consequently (0;b) is the point of intersection with the y-axis.
Note.If b = 0 and k = 0, then the function y=0 vanishes for any value of the variable X. If a b ≠ 0 and k = 0, then the function y=b does not vanish for any value of the variable X.
6) The intervals of constancy of sign depend on the coefficient k.
a) k > 0; kx + b > 0, kx > -b, x > -b/k.
y = kx + b- positive at x from (-b/k; +∞),
y = kx + b- negative at x from (-∞; -b/k).
b) k< 0; kx + b < 0, kx < -b, x < -b/k.
y = kx + b- positive at x from (-∞; -b/k),
y = kx + b- negative at x from (-b/k; +∞).
c) k = 0, b > 0; y = kx + b positive throughout the domain of definition,
k = 0, b< 0; y = kx + b is negative throughout the domain of definition.
7) Intervals of monotonicity of a linear function depend on the coefficient k.
k > 0, Consequently y = kx + b increases over the entire domain of definition,
k< 0 , Consequently y = kx + b decreases over the entire domain of definition.
8) The graph of a linear function is a straight line. To draw a straight line, it is enough to know two points. The position of the straight line on the coordinate plane depends on the values of the coefficients k and b. Below is a table that clearly illustrates this.
