X y is 2 graph of a linear function. Direct function

The concept of a numerical function. Ways to set a function. Function properties.

A numeric function is a function that acts from one number space (set) to another number space (set).

There are three main ways to define a function: analytical, tabular and graphical.

1. Analytical.

The method of specifying a function using a formula is called analytical. This method is the main one in the mat. analysis, but in practice it is not convenient.

2. Tabular way of setting the function.

A function can be defined using a table containing the argument values ​​and their corresponding function values.

3. Graphical way of setting the function.

The function y \u003d f (x) is called given graphically if its graph is built. This method of setting the function makes it possible to determine the values ​​of the function only approximately, since the construction of a graph and finding the values ​​of the function on it is associated with errors.

Properties of a function that must be taken into account when plotting its graph:

1) The scope of the function.

Function scope, that is, those values ​​that the argument x of the function F =y (x) can take.

2) Intervals of increasing and decreasing function.

The function is called increasing on the considered interval, if the larger value of the argument corresponds to the larger value of the function y(x). This means that if two arbitrary arguments x 1 and x 2 are taken from the interval under consideration, and x 1 > x 2, then y (x 1) > y (x 2).

The function is called decreasing on the interval under consideration, if the larger value of the argument corresponds to the smaller value of the function y(x). This means that if two arbitrary arguments x 1 and x 2 are taken from the considered interval, and x 1< х 2 , то у(х 1) < у(х 2).

3) Function zeros.

The points at which the function F \u003d y (x) intersects the abscissa axis (they are obtained by solving the equation y (x) \u003d 0) and are called the zeros of the function.

4) Even and odd functions.

The function is called even, if for all values ​​of the argument from the scope

y(-x) = y(x).

The graph of an even function is symmetrical about the y-axis.

The function is called odd, if for all values ​​of the argument from the scope

y(-x) = -y(x).

The graph of an even function is symmetrical with respect to the origin.

Many functions are neither even nor odd.

5) Periodicity of the function.

The function is called periodic, if there is a number P such that for all values ​​of the argument from the domain of definition

y(x + P) = y(x).

Linear function, its properties and graph.

A linear function is a function of the form y = kx + b, defined on the set of all real numbers.

k– slope factor (real number)

b– free term (real number)

x is an independent variable.

· In a particular case, if k = 0, we get a constant function y = b, the graph of which is a straight line parallel to the Ox axis, passing through the point with coordinates (0; b).

· If b = 0, then we get the function y = kx, which is a direct proportionality.

o The geometric meaning of the b coefficient is the length of the segment that the straight line cuts off along the Oy axis, counting from the origin.

o The geometric meaning of the coefficient k is the angle of inclination of the straight line to the positive direction of the Ox axis, it is considered counterclockwise.

Linear function properties:

1) The domain of definition of a linear function is the entire real axis;

2) If k ≠ 0, then the range of the linear function is the entire real axis.

If 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, therefore, y = b is even;

b) b = 0, k ≠ 0, hence y = kx is odd;

c) b ≠ 0, k ≠ 0, hence y = kx + b is a general function;

d) b = 0, k = 0, hence y = 0 is both an even and an odd function.

4) The linear function does not have the property of periodicity;

5) Intersection points with coordinate axes:

Ox: y \u003d kx + b \u003d 0, x \u003d -b / k, therefore (-b / k; 0) is the point of intersection with the abscissa axis.

Oy: y = 0k + b = b, therefore (0; b) is the point of intersection with the y-axis.

Comment. If b = 0 and k = 0, then the function y = 0 vanishes for any value of x. If b ≠ 0 and k = 0, then the function y = b does not vanish for any values ​​of the variable x.

6) Intervals of sign constancy depend on the coefficient k.

a) k > 0; kx + b > 0, kx > -b, x > -b/k.

y = kx + b is positive for x from (-b/k; +∞),

y = kx + b is negative for x from (-∞; -b/k).

b) k< 0; kx + b < 0, kx < -b, x < -b/k.

y = kx + b is positive for x from (-∞; -b/k),

y = kx + b is negative for x from (-b/k; +∞).

c) k = 0, b > 0; y = kx + b is positive throughout the domain,

k = 0, b< 0; y = kx + b отрицательна на всей области определения.

7) Intervals of monotonicity of a linear function depend on the coefficient k.

k > 0, hence y = kx + b increases over the entire domain,

k< 0, следовательно y = kx + b убывает на всей области определения.

11. Function y \u003d ax 2 + bx + c, its properties and graph.

The function y \u003d ax 2 + bx + c (a, b, c are constant values, a ≠ 0) is called quadratic. In the simplest case, y \u003d ax 2 (b \u003d c \u003d 0), the graph is a curved line passing through the origin. The curve serving as a graph of the function y \u003d ax 2 is a parabola. Every parabola has an axis of symmetry called axis of the parabola. The point O of the intersection of the parabola with its axis is called top of the parabola.

The graph can be built according to the following scheme: 1) Find the coordinates of the top of the parabola x 0 = -b/2a; y 0 \u003d y (x 0). 2) We build a few more points that belong to the parabola, when building, you can use the symmetries of the parabola with respect to the straight line x = -b / 2a. 3) We connect the indicated points with a smooth line. Example. Construct a graph of the function in \u003d x 2 + 2x - 3. Solutions. The graph of the function is a parabola whose branches are directed upwards. The abscissa of the top of the parabola x 0 \u003d 2 / (2 ∙ 1) \u003d -1, its ordinates y (-1) \u003d (1) 2 + 2 (-1) - 3 \u003d -4. So, the top of the parabola is the point (-1; -4). Let's make a table of values ​​for several points that are placed to the right of the axis of symmetry of the parabola - the straight line x \u003d -1. Function properties.

Consider the function y=k/y. The graph of this function is a line, called a hyperbola in mathematics. The general view of the hyperbola is shown in the figure below. (The graph shows a function y equals k divided by x, where k is equal to 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 comes closer and closer to the coordinate axes in one of the directions. The coordinate axes in this case are called asymptotes.

In general, any straight lines that the graph of a function infinitely approaches, but does not reach, are called asymptotes. A hyperbola, like a parabola, has axes of symmetry. For the hyperbola shown in the figure above, this is the straight line y=x.

Now let's deal with two general cases of hyperbolas. 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.

Main properties of the function y = k/x, for k>0

Graph of the function y = k/x, for k>0

5. y>0 for x>0; y6. The function decreases both on the interval (-∞;0) and on the interval (0;+∞).

10. The range of the function is two open intervals (-∞;0) and (0;+∞).

The main properties of the function y = k/x, for k<0

Graph of the function y = k/x, for k<0

1. The point (0;0) is the center of symmetry of the hyperbola.

2. Axes of coordinates - asymptotes of a hyperbola.

4. The scope of the function is all x, except x=0.

5. y>0 for x0.

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

7. The function is not limited from below or from above.

8. The function has neither the largest nor the smallest values.

9. The function is continuous on the interval (-∞;0) and on the interval (0;+∞). Has a gap at the point x=0.

"Critical points of the function" - Critical points. Among the critical points there are extremum points. A necessary condition for an extremum. Answer: 2. Definition. But, if f "(x0) = 0, then it is not necessary that the point x0 will be an extremum point. Extreme points (repetition). Critical points of the function. Extreme points.

"Coordinate plane Grade 6" - Mathematics Grade 6. 1. X. 1. Find and write down the coordinates of points A, B, C, D: -6. Coordinate plane. O. -3. 7. W.

"Functions and their graphs" - Continuity. The largest and smallest value of the function. The concept of an inverse function. Linear. Logarithmic. Monotone. If k > 0, then the formed angle is acute, 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 Grade 9" - Permissible arithmetic operations on functions. [+] - addition, [-] - subtraction, [*] - multiplication, [:] - division. In such cases, one speaks of a graphical specification of a function. Formation of a class of elementary functions. Power function y=x0.5. Iovlev Maxim Nikolaevich, a student of the 9th grade of the RIOU Raduzhskaya school.

"Lesson Tangent Equation" - 1. Clarify the concept of a tangent to a function graph. Leibniz considered the problem of drawing a tangent to an arbitrary curve. ALGORITHM FOR COMPOSING THE EQUATION OF THE FUNCTION tangent to the GRAPH y=f(x). Lesson topic: Test: find the derivative of a function. Tangent equation. Fluxion. Grade 10. Decipher how Isaac Newton called the derivative of a function.

"Build a graph of the function" - The function y=3cosx is given. Graph of the function y=m*sin x. Plot the function graph. Content: Given a function: y=sin (x+?/2). Stretching the graph y=cosx along the y axis. To continue press L. Mouse button. The function y=cosx+1 is given. Graph offsets y=sinx vertically. The function y=3sinx is given. Graph offset y=cosx horizontally.

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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 $k > 0$.

1. $f"\left(x\right)=(\left(kx+b\right))"=k>0$. Therefore, this function increases over the entire domain of definition. There are no extreme points.
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 scope is all numbers.
2. The scope is all numbers.
3. $f\left(-x\right)=-kx+b$. The function is neither even nor odd.
4. For $x=0,f\left(0\right)=b$. For $y=0,0=kx+b,\ x=-\frac(b)(k)$.

Intersection points with coordinate axes: $\left(-\frac(b)(k),0\right)$ and $\left(0,\ b\right)$

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 leaving town A this moment is located 20 km away. At what distance s (km) from A will the motorcyclist be after t hours if he moves at a speed of 40 km/h?

It is obvious that in t hours the motorcyclist will travel 50t km. Consequently, after t hours it 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 a 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 be paid for sending a telegram containing n words?

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

In both problems considered, we encountered functions that are given by formulas of the form y \u003d kx + l, where k and l are some numbers, and x and y are variables.

A function that can be given 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 a linear function can be the set of all numbers or any of its subsets.

A special case of a linear function is the previously considered direct proportionality. Recall that for l \u003d 0 and k ≠ 0, the formula y \u003d kx + l takes the form y \u003d kx, and this formula, as you know, for k ≠ 0, direct proportionality is given.

Let us need to plot a linear function f given by the formula
y \u003d 0.5x + 2.

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

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

It is obvious that the constructed points lie on some straight line. It does not yet follow from this that the graph of this function is a straight line.

To find out what form the graph of the considered function f has, let's compare it with the graph of direct proportionality x - y familiar to us, where x \u003d 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 of the graph of the function f is greater than the corresponding ordinate of the direct proportionality graph by 2 units.

Therefore, the graph of the considered function f 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 considered linear function f is also a straight line.

In general, the graph of a function given by a formula of the form y \u003d 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 \u003d 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 line through these points. This straight line is the desired graph.

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

The location of the graph of the function given by the formula y \u003d kx + l depends on the values ​​\u200b\u200bof l and k. In particular, the value of the angle of inclination of the graph of a linear function to the x-axis depends on the coefficient k. If k is a positive number, then this angle is acute; if k is a negative number, then the angle is obtuse. The number k is called the slope of the line.

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