Advice 1: How to find a function with the graphics

In school we studied in detail the functions and construct their graphs. However, to read the graph of the function and find its kind on the finished drawing, we are, unfortunately, practically is not taught. In fact, it is not difficult if you remember a few basic types of functions.The task of describing the properties of a function from its schedule often arises in experimental studies. For graphics , you can determine the intervals of increasing and decreasing functions, discontinuities and extrema, and also you can see the asymptotes.
How to find a function with the graphics
Instruction
1
If the graph is a straight line passing through the origin and forming with the axis OX the angle α (the angle of inclination of a straight line to the positive half-axis OX). The function that describes this line will have the form y = kx. The coefficient of proportionality k is equal to tg α. If the line passes through 2 nd and 4-th quadrant, k < 0, and the function is decreasing if in the 1st and 3rd, then k > 0 and a function increases.Let the graph is a straight line, arranged in different ways relative to the axes of coordinates. It is a linear function, and it has the form y = kx + b, where variables x and y are in the first degree, and k and b can take both positive and negative values or zero. Video is parallel to the line y = kx cuts and on the y-axis |b| units. A line parallel to the x-axis, then k = 0 if the y-axis, the equation has the form x = const.
2
A curve consisting of two branches, located in different quarters and symmetric about the origin, is called a hyperbola. This graph expresses the inverse relationship of the variable y from x and is described by the equation y = k/x. Here k ≠ 0 is the coefficient of inverse proportionality. In this case, if k > 0, the function decreases; if k < 0 the function is increasing. Thus, the domain of the function is the entire number line except x = 0. The branches of the hyperbola approach the coordinate axes as their asymptotes. With decreasing k the branches of the hyperbola are more "pressed" into the coordinate angles.
3
A quadratic function has the form y = ax2 + bx + C, where a, b and c values constant and a  0. If the condition b = C = 0, a function equation looks like y = ax2 (the simplest case of a quadratic function), and its graph is a parabola passing through the origin. The graph of the function y = ax2 + bx + C has the same form as the simplest case of functions, however, its peak (the point of intersection of the parabola with the axis OY) lies not in the origin.
4
A parabola is also the graph of the exponential function expressed by the equation y = xⁿ, if n is any even number. If n is any odd number, the graph of this exponential function will have the form of a cubic parabola.
If n is any negative number, a function equation takes the form. The graph of the function with odd n is a hyperbola, and if n is even their branches are symmetric about the axis Oy.

Advice 2: How to define a function according to the schedule

Coordinate of any point on the plane is determined by two variables: on the x-axis and the y-axis. A set of such points and represents the graph of the function. On it you see a change in the Y value depending on changing the value of X. you can Also determine in which area (gap) the function increases and what decreases.
How to define a function according to the schedule
Instruction
1
What can we say about a function if its graph is a straight line? Let's see if this video passes through the origin of coordinates (that is, one where the values of X and Y is equal to 0). If held, 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 y-axis will be located this video. And the Y-axis actually corresponds to an infinitely large value of k.
2
Look at the direction of the function. If it goes "lower left – top right", that is, the 3rd and the 1st quadrant, it is increasing, if "top left to right down" (via the 2nd and 4th quarters), it is waning.
3
When video does not pass through the origin, it is described by the equation y = kx + b. Video crosses the y-axis at the point where y = b and y value can be positive or negative.
4
A function is called a parabola, if is described by the equation y = x^n, and its appearance depends on the value of n. If n is any even number (the simplest case of a quadratic function y = x^2) function graph is a curve passing through the origin and through the point with coordinates (1;1), (-1;1), since the unit in any degree will remain one. All values of y corresponding to any value X other than zero, can only be positive. The function is symmetric about the Y-axis and its graph is 1-St and 2-nd coordinate quarters. It is easy to understand that the larger the value of n, the close graph is to the y-axis.
5
If n is an odd number, the graph of this function is a cubic parabola. Curve is located in the 1st and 3rd coordinate quadrants, symmetrical with respect to the Y-axis and passes through the origin and through the point (-1;-1), (1;1). When a quadratic function is an equation y = ax^2 + bx + c form of the parabola coincides with the form in the simplest case (y = x^2), but its peak is not at the point of origin.
6
A function is called a hyperbola, if it is described by the equation y = k/x. It is easy to see that when the value of x tends to 0, the value of y increases to infinity. The function graph is a curve consisting of two branches which are located in different coordinate quarters.
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