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 increases. Let it represent a straight line located in different 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 graph is the inverse dependence of the variable y on 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.
Quadratic function has the form y = ax2 + bx + c, where a, b and c are constant quantities and a 0. When the condition b = c = 0 is met, the equation of the function looks like y = ax2 (the 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.
Even in school years, 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 of 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 of a 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. If the variable n is any negative number, the function equation 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.
1) Function domain and function range.
The domain of a function is the set of all valid valid argument values x(variable x), for which the function y = f(x) determined. The range of a function is the set of all real values y, which the function accepts.
In elementary mathematics, functions are studied only on the set of real numbers.
2) Function zeros.
Function zero is the value of the argument at which the value of the function is equal to zero.
3) Intervals of constant sign of a function.
Intervals of constant sign of a function are sets of argument values on which the function values are only positive or only negative.
4) Monotonicity of the function.
An increasing function (in a certain interval) is a function for which higher value the argument from this interval corresponds to a larger value of the function.
A decreasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.
5) Even (odd) function.
An even function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality f(-x) = f(x). Schedule even function symmetrical about the ordinate axis.
An odd function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality is true f(-x) = - f(x). Schedule odd function symmetrical about the origin.
6) Limited and unlimited functions.
A function is called bounded if there is a positive number M such that |f(x)| ≤ M for all values of x. If such a number does not exist, then the function is unlimited.
7) Periodicity of the function.
A function f(x) is periodic if there is a non-zero number T such that for any x from the domain of definition of the function the following holds: f(x+T) = f(x). This smallest number is called the period of the function. All trigonometric functions are periodic. (Trigonometric formulas).
19. Basic elementary functions, their properties and graphs. Application of functions in economics.
Basic elementary functions. Their properties and graphs
1. Linear function.
Linear function is called a function of the form , where x is a variable, a and b are real numbers.
Number A called the slope of the line, it is equal to the tangent of the angle of inclination of this line to the positive direction of the x-axis. The graph of a linear function is a straight line. It is defined by two points.
Properties of a Linear Function
1. Domain of definition - the set of all real numbers: D(y)=R
2. The set of values is the set of all real numbers: E(y)=R
3. The function takes a zero value when or.
4. The function increases (decreases) over the entire domain of definition.
5. Linear function continuous over the entire domain of definition, differentiable and .
2. Quadratic function.
A function of the form, where x is a variable, coefficients a, b, c are real numbers, is called quadratic
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 $k$. Slope factor 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 $k > 0$.
- $f"\left(x\right)=(\left(kx+b\right))"=k>0$. 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).
