Properties of linear function coefficients. GIA. Quadratic function

"Critical points of the function" - critical points. Among critical points there are points of extremum. Required extremma condition. Answer: 2. Definition. But, if f "(x0) \u003d 0, it is not necessary that the point x0 is an extremum point. Extremum points (repetition). Critical points function of the point of extremums.

"Coordinate Plane Grade 6" - Mathematics Grade 6. 1. H. 1.Nate and record the coordinates of the points A, B, C, D: -6. Coordinate plane. O. -3. 7. U.

"Functions and their graphs" - continuity. The greatest and smallest function of the function. The concept of reverse function. Linear. Logarithmic. Monotone. If k\u003e 0, then the angle is sharp, if k< 0, то угол тупой. В самой точке x = a функция может существовать, а может и не существовать. Х1, х2, х3 – нули функции у = f(x).

"Functions 9 Class" - permissible arithmetic actions on functions. [+] - Addition, [-] - subtraction, [*] - multiplication, [:] - division. In such cases, they talk about the graphical task of the function. Education class of elementary functions. The power function y \u003d x0.5. Iovleva Maxim Nikolayevich, student 9 grade RMOU Raduzhskaya Oosh.

"Lesson Equation of tangent" - 1. To clarify the concept of tangent to the graphics of the function. Leibniz considered the task of conducting a tangential curve. The algorithm for the preparation of the equation tangent to the graph of the function y \u003d f (x). Topic lesson: Test: Find a derived function. Equation tangent. Fluxion. Grade 10. Decipher how Isaac Newton called a derivative function.

"Build a function graph" - the Y \u003d 3COSX function is given. Function graph Y \u003d M * SIN x. Build a graph of the function. Content: Dana function: y \u003d sin (x +? / 2). Stretching graph Y \u003d COSX along the Y axis. To continue click on l. Mouse button. The function y \u003d cosx + 1 is given. Shooting the graph y \u003d sinx vertically. The function y \u003d 3sinx is given. Shooting the graph Y \u003d COSX horizontally.

Total in the subject of 25 presentations

Instruction

If the schedule is a straight line passing through the origin of the coordinates and an angle of α (the angle of the straight to the positive semi-axis oh). A function describing this direct will be viewed y \u003d kx. The ratio of the proportionality K is Tg α. If direct 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 increases. Pause is a straight line, which is in different ways relative to the axes of coordinates. This is a linear function, and it has the form y \u003d kx + b, where the variables x and y are in the first degree, and k and b can receive both positive and negative values \u200b\u200bor zero. Direct parallel direct Y \u003d KX and cuts off on the axis | B | units. If the straight is parallel to the abscissa axis, then k \u003d 0, if the axis is ordinate, the equation has the form x \u003d const.

A curve consisting of two branches located in different quarters and symmetrical relative to the origin of the coordinates, hyperbole. This graph is the inverse dependence of the variable Y from x and is described by the Y \u003d k / x equation. Here k ≠ 0 is the proportionality coefficient. In this case, if k\u003e 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 \u003d ax2 + bx + C, where a, b and c - permanent values \u200b\u200band a  0. When the condition is performed b \u003d C \u003d 0, the function equation looks like Y \u003d AX2 (the simplest case), and its The schedule is a parabola passing through the origin of the coordinates. The graph of the function y \u003d ax2 + BX + C has the same form as the simplest case of the function, however, its vertex (the intersection point with the Oy axis) is not at the beginning of the coordinates.

The parabola is also a graph of a powerful function expressed by the equation y \u003d xⁿ if N is any even number. If N is any odd number, the graph of such a power function will have a kind of cubic parabola.
In case N - any, the function equation acquires the view. The graph of the function with an odd n will be hyperbole, and with even ns their branches will be symmetrical relative to the OU axis.

Back in school years, the functions are studied in detail and their graphics are built. But, unfortunately, read the graph of the function and find its type on the presented drawing is practically not taught. In fact, it is quite simple if you remember the main types of functions.

Instruction

If the represented schedule is, which through the origin of the coordinates and with the OX axis angle α (which is an angle of inclination direct to the positive semi-axis), then the function describing this direct will be presented as Y \u003d KX. In this case, the proportionality K is equal to the tangent of the angle α.

If the specified straight line passes through the second and fourth coordinate quarters, then K is 0, and the function increases. Let the presented schedule be a straight line, located in any way relative to the axes of coordinates. Then the function of this graphics It will be linear, which is represented by the type y \u003d kx + b, where the variables y and x stand in the first, and B and K can take both negative and positive values \u200b\u200bor.

If direct is parallel to the straight line with the Y \u003d KX graph and cuts out on the axis of the ordinate b units, then the equation has the form x \u003d const if the graph is parallel to the abscissa axis, then k \u003d 0.

The curve line, which consists of two branches, symmetrical about the origin of the coordinates and are located in different quarters, hyperbole. Such a graph shows the inverse dependence of the variable Y from the variable x and is described by the equation of the form y \u003d k / x, where k should not be zero, since it is a coefficient of reverse proportionality. In this case, if the value k is greater than zero, the function decreases; If K is less than zero - increases.

If the proposed schedule is a parabola passing through the origin of the coordinates, its function when performing the condition that B \u003d C \u003d 0, will have the form Y \u003d AX2. This is the easiest case of a quadratic function. The graph of the function of the type y \u003d ax2 + BX + C will have the same appearance as the simplest case, however, the top (point where the schedule intersects with the ordinate axis) will not be at the beginning of the coordinates. In the quadratic function, represented by the type y \u003d Ax2 + BX + C, the values \u200b\u200bof the values \u200b\u200bof A, B and C are constant, with no equally zero.

A parabola can also be a graph of a powerful function, a pronounced equation of the form y \u003d Xⁿ, only if N is any even number. If the value n is an odd number, such a graph of the power function will be represented by cubic parabola. In case the variable N is any negative number, the function equation acquires the view.

Video on the topic

The coordinate of absolutely any point on the plane is determined by two its values: along the abscissa axis and the ordinate axis. A combination of many such points and represents a graph of a function. According to him, you see how the value of y is changing depending on the change in the value of X. Also you can determine on which site (gap) the function increases, and what decreases.

Instruction

What can be said about the function if its schedule is a straight line? Look, whether this straight line passes through the point of origin of the coordinate (that is, the one where the values \u200b\u200bX and Y are equal to 0). If it passes, this function is described by the Y \u003d KX equation. It is easy to understand that the greater the value of K, the closer to the axis the ordinate will be located this straight. And the Y axis itself actually corresponds to an infinitely large value of k.

The tasks for properties and graphs of the quadratic function cause, as practice shows, serious difficulties. It is rather strange, because the quadratic function is held in the 8th grade, and then the entire first quarter of the 9th grade "survive" the properties of the parabola and build its graphs for various parameters.

This is due to the fact that forcing students to build parabolas, almost do not pay time for reading charts, that is, not practicing the understanding of the information obtained from the picture. Apparently, it is assumed that by building a dozen two charts, a smart schoolboy will independently discover and formulate the relationship of coefficients in the formula and the appearance of the graph. In practice it does not work. For such a generalization, a serious experience of mathematical mini studies, which most nine-graduates, of course, do not have it. Meanwhile, in GIA suggest precisely on the schedule to determine the signs of coefficients.

Let's not require schoolchildren impossible and simply offer one of the algorithms to solve such problems.

So, the function of the form y \u003d AX 2 + BX + C It is called a quadratic, the schedule is parabola. As follows from the name, the main term is aX 2.. I.e but should not be zero, the remaining coefficients ( b. and from) can be zero.

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

The simplest dependence for the coefficient but. Most schoolchildren confidently replies: "If but \u003e 0, then the parabola branches are directed upwards, and if but < 0, - то вниз". Совершенно верно. Ниже приведен график квадратичной функции, у которой but > 0.

y \u003d 0.5X 2 - 3X + 1

In this case but = 0,5

And now for but < 0:

y \u003d - 0.5x2 - 3x + 1

In this case but = - 0,5

Influence of the coefficient from Also easy to trace enough. Imagine that we want to find the value of the function at the point h. \u003d 0. Substitute zero in the formula:

y. = a. 0 2 + b. 0 + c. = c.. Turns out that y \u003d s. I.e from - This is the ordinate of the point of intersection of the parabola with the axis. As a rule, this point is easy to find on the chart. And determine above zero it lies or below. I.e from \u003e 0 or from < 0.

from > 0:

y \u003d x 2 + 4x + 3

from < 0

y \u003d x 2 + 4x - 3

Accordingly, if from \u003d 0, then Parabola will definitely pass through the origin of the coordinate:

y \u003d x 2 + 4x

More difficult with the parameter b.. The point on which we will find it depends not only from b. But from but. This is the top of the parabola. Its abscissa (axis coordinate h.) is on the formula x B \u003d - b / (2a). In this way, b \u003d - 2ach in. That is, we act as follows: on the chart we find the top of the parabola, we define the sign of its abscissa, that is, we look to the right of zero ( x B. \u003e 0) or left ( x B. < 0) она лежит.

However, this is not all. We also need to pay attention to the coefficient sign but. That is, to see where the branches of parabola are directed. And only after that by the formula b \u003d - 2ach in Determine the sign b..

Consider an example:

Branches are directed up, it means but \u003e 0, Parabola crosses the axis w. below zero, then from < 0, вершина параболы лежит правее нуля. Следовательно, x B. \u003e 0. So b \u003d - 2ach in = -++ = -. b. < 0. Окончательно имеем: but > 0, b. < 0, from < 0.

Consider the function y \u003d k / y. The graph of this feature is a line called in math hyperbole. General view of hyperboles, presented in the figure below. (The graph shows the function y equal to K divided by x, in which K is equal to one.)

It can be seen that the schedule consists of two parts. These parts refer to the branches of hyperboles. It is also worth noting that each branch of hyperboles is suitable in one of the directions and closer and closer to the coordinate axes. The axis of the coordinates in this case is called asymptotes.

In general, any straight lines to which the chart of the function is infinitely approaching, but does not reach them, are called asymptotes. Hyperbolas, like a parabola, there are axis of symmetry. For hyperboles shown in the figure above, it is straight y \u003d x.

Now we'll figure it out with two general cases of hyperball. The graph of the function y \u003d k / x, for k ≠ 0, will be hyperbole, the branches of which are located either in the first and third coordinate corners, at k\u003e 0, or in the second and fourth coordinate angles, when<0.

The main properties of the function y \u003d k / x, when k\u003e 0

Function graph Y \u003d k / x, when k\u003e 0

5. Y\u003e 0 for x\u003e 0; y6. The function decreases both on the interval (-∞; 0) and in the interval (0; + ∞).

10. Area of \u200b\u200bfunction values \u200b\u200bTwo open gaps (-∞; 0) and (0; + ∞).

The main properties of the function y \u003d k / x, when k<0

Function graph Y \u003d k / x, when k<0

1. Point (0; 0) Center of symmetry hyperboles.

2. The axes of the coordinates - asymptotes of hyperboles.

4. The field of determining the function of all x, except x \u003d 0.

5. Y\u003e 0 at x0.

6. The function increases both on the interval (-∞; 0) and on the interval (0; + ∞).

7. The function is not limited to the bottom, none.

8. The function has no greatest nor the least values.

9. The function is continuous on the interval (-∞; 0) and on the interval (0; + ∞). It has a gap at point x \u003d 0.

Definition of linear function

We introduce the definition of a linear function

Definition

The function of the type $ y \u003d kx + b $, where $ k $ is different from zero called a linear function.

The graph of the linear function is straight. The number $ k $ is called the corner coefficient of direct.

For $ b \u003d 0 $, the linear function is called the function of direct proportionality $ y \u003d kx $.

Consider Figure 1.

Fig. 1. Geometric meaning of the angular coefficient of direct

Consider the triangle ABC. We see that $ aircraft \u003d kx_0 + b $. We will find the intersection point direct $ Y \u003d KX + B $ with an axis $ OX $:

\ \

So $ AC \u003d X_0 + \\ FRAC (B) (K) $. Find the attitude of these parties:

\\ [\\ FRAC (BC) (AC) \u003d \\ FRAC (KX_0 + B) (X_0 + \\ FRAC (B) (K)) \u003d \\ FRAC (K (KX_0 + B)) ((kx) _0 + b) \u003d k \\]

On the other hand $ \\ FRAC (BC) (AC) \u003d TG \\ Angle A $.

Thus, you can draw the following conclusion:

Output

Geometric meaning of the coefficient of $ k $. The corner coefficient of direct $ k $ is equal to the tangent angle of tilt this direct to the $ ox $ axis.

Study of the linear function $ F \\ left (x \\ right) \u003d KX + B $ and its schedule

First, consider the function $ F \\ left (x \\ right) \u003d kx + b $, where $ k\u003e 0 $.

$ f "\\ left (x \\ right) \u003d (\\ left (kx + b \\ right))" \u003d k\u003e 0 $. Consequently, this function increases throughout the field of definition. Points of extremum is not.
$ (\\ Mathop (LIM) _ (x \\ to - \\ infty) kx \\) \u003d - \\ infty $, $ (\\ mathop (lim) _ (x \\ to + \\ infty) kx \\) \u003d + \\ infty $
Graph (Fig. 2).

Fig. 2. The graphs of the function $ y \u003d kx + b $, with $ k\u003e 0 $.

Now consider the function $ F \\ Left (X \\ Right) \u003d KX $, where $ k

The definition area is all numbers.
Value area - all numbers.
$ F \\ Left (-x \\ Right) \u003d - KX + B $. The function is neither even nor odd.
At $ x \u003d 0, f \\ left (0 \\ right) \u003d b $. For $ y \u003d 0.0 \u003d kx + b, \\ x \u003d - \\ FRAC (b) (k) $.

Point of intersection with axes of coordinates: $ \\ left (- \\ FRAC (B) (K), 0 \\ Right) $ and $ \\ left (0, \\ b \\ right) $

$ f "\\ left (x \\ right) \u003d (\\ left (kx \\ right))" \u003d k
$ F ^ ("") \\ left (x \\ right) \u003d k "\u003d 0 $. Therefore, the function does not have the flexion points.
$ (\\ Mathop (Lim) _ (x \\ to - \\ infty) kx \\) \u003d + \\ infty $, $ (\\ mathop (lim) _ (x \\ to + \\ infty) kx \\) \u003d - \\ infty $
Graph (Fig. 3).