Consider the function y=k/y. The graph of this function is a line, called a hyperbola in mathematics. The general view of a hyperbola is shown in the figure below. (The graph shows the function y equals k divided by x, for which k equals 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 approaches in one of the directions closer and closer to the coordinate axes. The coordinate axes in this case are called asymptotes.
In general, any straight lines to which the graph of a function infinitely approaches but does not reach them are called asymptotes. A hyperbola, like a parabola, has axes of symmetry. For the hyperbola shown in the figure above, this is the line y=x.
Now let's look at two common cases of hyperbole. 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.
Basic properties of the function y = k/x, for k>0
Graph of the function y = k/x, for k>0
5. y>0 at x>0; y6. The function decreases both on the interval (-∞;0) and on the interval (0;+∞).
10. The range of values of the function is two open intervals (-∞;0) and (0;+∞).
Basic properties of the function y = k/x, for k<0
Graph of the function y = k/x, at k<0
1. Point (0;0) is the center of symmetry of the hyperbola.
2. Coordinate axes - asymptotes of the hyperbola.
4. The domain of definition of the function is all x except x=0.
5. y>0 at x0.
6. The function increases both on the interval (-∞;0) and on the interval (0;+∞).
7. The function is not limited either from below or from above.
8. A function has neither a maximum nor a minimum value.
9. The function is continuous on the interval (-∞;0) and on the interval (0;+∞). Has a gap at x=0.
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 the 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 coefficient needs to 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.
Take the derivative of the function given to you. There is no need to build a graph here - you only need the equation of the function. In our example, take the derivative of the function. Take the derivative according to the methods outlined in the article mentioned above:
- Derivative:
Substitute the coordinates of the point given to you into the found derivative to calculate the slope. The derivative of a function is equal to the slope at a certain point. In other words, f"(x) is the slope of the function at any point (x,f(x)). In our example:
- Find the slope of the function f (x) = 2 x 2 + 6 x (\displaystyle f(x)=2x^(2)+6x) at point A(4,2).
- Derivative of a function:
- f ′ (x) = 4 x + 6 (\displaystyle f"(x)=4x+6)
- Substitute the value of the “x” coordinate of this point:
- f ′ (x) = 4 (4) + 6 (\displaystyle f"(x)=4(4)+6)
- Find the slope:
- Slope function f (x) = 2 x 2 + 6 x (\displaystyle f(x)=2x^(2)+6x) at point A(4,2) is equal to 22.
If possible, check your answer on a graph. Remember that the slope cannot be calculated at every point. Differential calculus deals with complex functions and complex graphs where the slope cannot be calculated at every point, and in some cases the points do not lie on the graphs at all. If possible, use a graphing calculator to check that the slope of the function you are given is correct. Otherwise, draw a tangent to the graph at the point given to you and think about whether the slope value you found matches what you see on the graph.
- The tangent will have the same slope as the graph of the function at a certain point. To draw a tangent at a given point, move left/right on the X axis (in our example, 22 values to the right), and then up one on the Y axis. Mark the point, and then connect it to the point given to you. In our example, connect the points with coordinates (4,2) and (26,3).
A linear function is a function of the form
x-argument (independent variable),
y-function (dependent variable),
k and b are some constant numbers
The graph of a linear function is straight.
To create a graph it is enough two points, because through two points you can draw a straight line and, moreover, only one.
If k˃0, then the graph is located in the 1st and 3rd coordinate quarters. If k˂0, then the graph is located in the 2nd and 4th coordinate quarters.
The number k is called the slope of the straight graph of the function y(x)=kx+b. If k˃0, then the angle of inclination of the straight line y(x)= kx+b to the positive direction Ox is acute; if k˂0, then this angle is obtuse.
Coefficient b shows the point of intersection of the graph with the op-amp axis (0; b).
y(x)=k∙x-- a special case of a typical function is called direct proportionality. The graph is a straight line passing through the origin, so one point is enough to construct this graph.
Graph of a Linear Function
Where coefficient k = 3, therefore
The graph of the function will increase and have an acute angle with the Ox axis because coefficient k has a plus sign.
OOF linear function
OPF of a linear function
Except in the case where
Also a linear function of the form
Is a function of general form.
B) If k=0; b≠0,
In this case, the graph is a straight line parallel to the Ox axis and passing through the point (0; b).
B) If k≠0; b≠0, then the linear function has the form y(x)=k∙x+b.
Example 1 . Graph the function y(x)= -2x+5
Example 2 . Let's find the zeros of the function y=3x+1, y=0;
– zeros of the function.
Answer: or (;0)
Example 3 . Determine the value of the function y=-x+3 for x=1 and x=-1
y(-1)=-(-1)+3=1+3=4
Answer: y_1=2; y_2=4.
Example 4 . Determine the coordinates of their intersection point or prove that the graphs do not intersect. Let the functions y 1 =10∙x-8 and y 2 =-3∙x+5 be given.
If the graphs of functions intersect, then the values of the functions at this point are equal
Substitute x=1, then y 1 (1)=10∙1-8=2.
Comment. You can also substitute the resulting value of the argument into the function y 2 =-3∙x+5, then we get the same answer y 2 (1)=-3∙1+5=2.
y=2- ordinate of the intersection point.
(1;2) - the point of intersection of the graphs of the functions y=10x-8 and y=-3x+5.
Answer: (1;2)
Example 5 .
Construct graphs of the functions y 1 (x)= x+3 and y 2 (x)= x-1.
You can see that the coefficient k=1 for both functions.
From the above it follows that if the coefficients of a linear function are equal, then their graphs in the coordinate system are located parallel.
Example 6 .
Let's build two graphs of the function.
The first graph has the formula
The second graph has the formula
In this case, we have a graph of two lines intersecting at the point (0;4). This means that the coefficient b, which is responsible for the height of the graph’s rise above the Ox axis, if x = 0. This means we can assume that the b coefficient of both graphs is equal to 4.
Editors: Ageeva Lyubov Aleksandrovna, Gavrilina Anna Viktorovna
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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 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.
The quadratic function has the form y = ax2 + bx + c, where a, b and c are constant quantities and a 0. If the condition b = c = 0 is met, the equation of the function looks like y = ax2 (the simplest case), and its the 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 the function, but its vertex (the point of intersection with the OY axis) does not lie at the origin.
A parabola is also the graph of a 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 graphics 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 to an infinitely large value of k.
