Function y kx b its properties. Linear function and its graph

Properties and graphs tasks quadratic function cause, as practice shows, serious difficulties. This is quite strange, because they study the quadratic function in the 8th grade, and then throughout the first quarter of the 9th grade they “torment” the properties of the parabola and build its graphs for various parameters.

This is due to the fact that when forcing students to construct parabolas, they practically do not devote time to “reading” the graphs, that is, they do not practice comprehending the information received from the picture. Apparently, it is assumed that, after constructing a dozen or two graphs, a smart student himself will discover and formulate the relationship between the coefficients in the formula and appearance graphic arts. In practice this does not work. For such a generalization, serious experience in mathematical mini-research is required, which most ninth-graders, of course, do not possess. Meanwhile, the State Inspectorate proposes to determine the signs of the coefficients using the schedule.

We will not demand the impossible from schoolchildren and will simply offer one of the algorithms for solving such problems.

So, a function of the form y = ax 2 + bx + c called quadratic, its graph is a parabola. As the name suggests, the main term is ax 2. That is A should not be equal to zero, the remaining coefficients ( b And With) can equal zero.

Let's see how the signs of its coefficients affect the appearance of a parabola.

The simplest dependence for the coefficient A. Most schoolchildren confidently answer: “if A> 0, then the branches of the parabola are directed upward, and if A < 0, - то вниз". Совершенно верно. Ниже приведен график квадратичной функции, у которой A > 0.

y = 0.5x 2 - 3x + 1

IN in this case A = 0,5

And now for A < 0:

y = - 0.5x2 - 3x + 1

In this case A = - 0,5

Impact of the coefficient With It's also pretty easy to follow. Let's imagine that we want to find the value of a function at a point X= 0. Substitute zero into the formula:

y = a 0 2 + b 0 + c = c. It turns out that y = c. That is With is the ordinate of the point of intersection of the parabola with the y-axis. Typically, this point is easy to find on the graph. And determine whether it lies above zero or below. That is With> 0 or With < 0.

With > 0:

y = x 2 + 4x + 3

With < 0

y = x 2 + 4x - 3

Accordingly, if With= 0, then the parabola will necessarily pass through the origin:

y = x 2 + 4x

More difficult with the parameter b. The point at which we will find it depends not only on b but also from A. This is the top of the parabola. Its abscissa (axis coordinate X) is found by the formula x in = - b/(2a). Thus, b = - 2ax in. That is, we proceed as follows: we find the vertex of the parabola on the graph, determine the sign of its abscissa, that is, we look to the right of zero ( x in> 0) or to the left ( x in < 0) она лежит.

However, that's not all. We also need to pay attention to the sign of the coefficient A. That is, look at where the branches of the parabola are directed. And only after that, according to the formula b = - 2ax in determine the sign b.

Let's look at an example:

The branches are directed upwards, which means A> 0, the parabola intersects the axis at below zero, that is With < 0, вершина параболы лежит правее нуля. Следовательно, x in> 0. So b = - 2ax in = -++ = -. b < 0. Окончательно имеем: A > 0, b < 0, With < 0.

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

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 $k$. The angular coefficient of the straight line $k$ is equal to the tangent of 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 $k > 0$.

1. $f"\left(x\right)=(\left(kx+b\right))"=k>0$. Hence, this function increases throughout the entire domain of definition. There are no extreme points.
2. $(\mathop(lim)_(x\to -\infty ) kx\ )=-\infty $, $(\mathop(lim)_(x\to +\infty ) kx\ )=+\infty $
3. 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

1. The domain of definition is all numbers.
2. The range of values ​​is all numbers.
3. $f\left(-x\right)=-kx+b$. The function is neither even nor odd.
4. 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 $\left(0,\ b\right)$

1. $f"\left(x\right)=(\left(kx\right))"=k
2. $f^("")\left(x\right)=k"=0$. Therefore, the function has no inflection points.
3. $(\mathop(lim)_(x\to -\infty ) kx\ )=+\infty $, $(\mathop(lim)_(x\to +\infty ) kx\ )=-\infty $
4. Graph (Fig. 3).

Let's consider the problem. A motorcyclist who left city A to currently is located 20 km from it. At what distance s (km) from A will the motorcyclist be after t hours if he moves at a speed of 40 km/h?

Obviously, in t hours the motorcyclist will travel 50t km. Consequently, after t hours he will be at a distance of (20 + 50t) km from A, i.e. s = 50t + 20, where t ≥ 0.

Each value of t corresponds to a single value of s.

The formula s = 50t + 20, where t ≥ 0, defines the function.

Let's consider one more problem. For sending a telegram, a fee of 3 kopecks is charged for each word and an additional 10 kopecks. How many kopecks (u) should you pay for sending a telegram containing n words?

Since the sender must pay 3n kopecks for n words, the cost of sending a telegram of n words can be found using the formula u = 3n + 10, where n is any natural number.

In both considered problems, we encountered functions that are given by formulas of the form y = kx + l, where k and l are some numbers, and x and y are variables.

A function that can be specified by a formula of the form y = kx + l, where k and l are some numbers, is called linear.

Since the expression kx + l makes sense for any x, the domain of definition of a linear function can be the set of all numbers or any subset of it.

A special case of a linear function is the previously discussed direct proportionality. Recall that for l = 0 and k ≠ 0 the formula y = kx + l takes the form y = kx, and this formula, as is known, for k ≠ 0 specifies direct proportionality.

Let us need to plot a linear function f given by the formula
y = 0.5x + 2.

Let's get several corresponding values ​​of the variable y for some values ​​of x:

Let's mark the points with the coordinates we received: (-6; -1), (-4; 0); (-2; 1), (0; 2), (2; 3), (4; 4); (6; 5), (8; 6).

Obviously, the constructed points lie on a certain line. It does not follow from this that the graph of this function is a straight line.

To find out what form the graph of the function f under consideration looks like, let’s compare it with the familiar graph of direct proportionality x – y, where x = 0.5.

For any x, the value of the expression 0.5x + 2 is greater than the corresponding value of the expression 0.5x by 2 units. Therefore, the ordinate of each point on the graph of the function f is 2 units greater than the corresponding ordinate on the graph of direct proportionality.

Consequently, the graph of the function f in question can be obtained from the graph of direct proportionality by parallel translation by 2 units in the direction of the y-axis.

Since the graph of direct proportionality is a straight line, then the graph of the linear function f under consideration is also a straight line.

In general, the graph of a function given by a formula of the form y = kx + l is a straight line.

We know that to construct a straight line it is enough to determine the position of its two points.

Let, for example, you need to plot a function that is given by the formula
y = 1.5x – 3.

Let's take two arbitrary values ​​of x, for example, x 1 = 0 and x 2 = 4. Calculate the corresponding values ​​of the function y 1 = -3, y 2 = 3, construct points A (-3; 0) and B (4; 3) and draw a straight line through these points. This straight line is the desired graph.

If the domain of definition of a linear function is not fully represented numbers, then its graph will be a subset of points on a line (for example, a ray, a segment, a set of individual points).

The location of the graph of the function specified by the formula y = kx + l depends on the values ​​of l and k. In particular, the angle of inclination of the graph of a linear function to the x-axis depends on the coefficient k. If k – positive number, then this angle is acute; if k – a negative number, then the angle is obtuse. The number k is called the slope of the line.

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Learn to take derivatives of functions. The derivative characterizes the rate of change of a function at a certain point lying on the graph of this function. In this case, the graph can be either a straight or curved line. That is, the derivative characterizes the rate of change of a function at a specific point in time. Remember general rules, by which derivatives are taken, and only then proceed to the next step.

• Read the article.
• How to take the simplest derivatives, for example, the derivative of an exponential equation, is described. The calculations presented in the following steps will be based on the methods described therein.

Learn to distinguish problems in which the slope must be calculated through the derivative of a function. Problems do not always ask you to find the slope or derivative of a function. For example, you may be asked to find the rate of change of a function at point A(x,y). You may also be asked to find the slope of the tangent at point A(x,y). In both cases it is necessary to take the derivative of the function.

>>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 the indicated transformations into general view.

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 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 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 practically a tourist cannot walk at a constant speed without sleep and rest for any amount of time . 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 construct points (-3; 7) and (2; -3) on the xOy coordinate plane 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 construct points (0; 4) and (6; 7) on the xOy coordinate plane 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 that the ordinates of the corresponding points of the straight 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

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 considered special cases of linear functions and learned what the relative position of graphs of linear functions depends on. 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 of knowledge in the application of linear functional dependence can be listed for a long time. Starting from phone charges, hair length and growth, and even proverbs in literature. And this list goes on and on.

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