Conduct a study of the function and construct its graph. Full examination of the function and plotting of the graph
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To fully study the function and plot its graph, it is recommended to use the following scheme:
1) find the domain of definition of the function;
2) find the discontinuity points of the function and vertical asymptotes (if they exist);
3) investigate the behavior of the function at infinity, find horizontal and oblique asymptotes;
4) examine the function for parity (oddness) and periodicity (for trigonometric functions);
5) find extrema and intervals of monotonicity of the function;
6) determine the convexity intervals and inflection points;
7) find the points of intersection with the coordinate axes, and, if possible, some additional points that clarify the graph.
The study of the function is carried out simultaneously with the construction of its graph.
Example 9 Explore the function and build a graph.
1. Scope of definition: ;
2. The function suffers discontinuity at points 
,
;
We examine the function for the presence of vertical asymptotes.
;
,
─ vertical asymptote.
;
,
─ vertical asymptote.
3. We examine the function for the presence of oblique and horizontal asymptotes.
Straight 
─ oblique asymptote, if 
,
.
,
.
Straight 
─ horizontal asymptote.
4. The function is even because 
. The parity of the function indicates the symmetry of the graph relative to the ordinate.
5. Find the monotonicity intervals and extrema of the function.
Let's find the critical points, i.e. points at which the derivative is 0 or does not exist: 
;
. We have three points 
;
. These points divide the entire real axis into four intervals. Let's define the signs 
on each of them.
On the intervals (-∞; -1) and (-1; 0) the function increases, on the intervals (0; 1) and (1; +∞) ─ it decreases. When passing through a point 
the derivative changes sign from plus to minus, therefore, at this point the function has a maximum 
.
6. Find the intervals of convexity and inflection points.
Let's find the points at which 
is 0, or does not exist.
has no real roots. 
,
,
Points 
And 
divide the real axis into three intervals. Let's define the sign 
at every interval.
Thus, the curve on the intervals 
And 
convex downwards, on the interval (-1;1) convex upwards; there are no inflection points, since the function is at points 
And 
not determined.
7. Find the points of intersection with the axes.
With axle 
the graph of the function intersects at the point (0; -1), and with the axis 
the graph does not intersect, because the numerator of this function has no real roots.
The graph of the given function is shown in Figure 1.
Figure 1 ─ Function graph 
Application of the concept of derivative in economics. Elasticity function
To study economic processes and solve other applied problems, the concept of elasticity of a function is often used.
Definition. Elasticity function 
is called the limit of the ratio of the relative increment of the function 
to the relative increment of the variable 
at 
, . (VII)
The elasticity of a function shows approximately how many percent the function will change 
when the independent variable changes 
by 1%.
The elasticity function is used in the analysis of demand and consumption. If the elasticity of demand (in absolute value) 
, then demand is considered elastic if 
─ neutral if 
─ inelastic relative to price (or income).
Example 10 Calculate the elasticity of the function 
and find the value of the elasticity index for 
= 3.
Solution: according to formula (VII), the elasticity of the function is:
Let x=3, then 
.This means that if the independent variable increases by 1%, then the value of the dependent variable will increase by 1.42%.
Example 11 Let the demand function 
regarding price 
looks like 
, Where 
─ constant coefficient. Find the value of the elasticity indicator of the demand function at price x = 3 den. units
Solution: calculate the elasticity of the demand function using formula (VII)
Believing 
monetary units, we get 
. This means that at a price 
monetary units a 1% increase in price will cause a 6% decrease in demand, i.e. demand is elastic.
Today we invite you to explore and build a graph of a function with us. After carefully studying this article, you will not have to sweat for long to complete this type of task. It is not easy to study and construct a graph of a function; it is a voluminous work that requires maximum attention and accuracy of calculations. To make the material easier to understand, we will study the same function step by step and explain all our actions and calculations. Welcome to the amazing and fascinating world mathematics! Go!
Domain
In order to explore and graph a function, you need to know several definitions. Function is one of the main (basic) concepts in mathematics. It reflects the dependence between several variables (two, three or more) during changes. The function also shows the dependence of sets.
Imagine that we have two variables that have a certain range of change. So, y is a function of x, provided that each value of the second variable corresponds to one value of the second. In this case, the variable y is dependent, and it is called a function. It is customary to say that the variables x and y are in For greater clarity of this dependence, a graph of the function is built. What is a graph of a function? This is a set of points on the coordinate plane, where each x value corresponds to one y value. Graphs can be different - straight line, hyperbola, parabola, sine wave, and so on.
It is impossible to graph a function without research. Today we will learn how to conduct research and build a graph of a function. It is very important to take notes during the study. This will make the task much easier to cope with. The most convenient research plan:
- Domain.
- Continuity.
- Even or odd.
- Periodicity.
- Asymptotes.
- Zeros.
- Sign constancy.
- Increasing and decreasing.
- Extremes.
- Convexity and concavity.
Let's start with the first point. Let's find the domain of definition, that is, on what intervals our function exists: y=1/3(x^3-14x^2+49x-36). In our case, the function exists for any values of x, that is, the domain of definition is equal to R. This can be written as follows xÎR.
Continuity
Now we will examine the discontinuity function. In mathematics, the term “continuity” appeared as a result of the study of the laws of motion. What is infinite? Space, time, some dependencies (an example is the dependence of the variables S and t in movement problems), the temperature of a heated object (water, frying pan, thermometer, etc.), a continuous line (that is, one that can be drawn without lifting it from the sheet pencil).
A graph is considered continuous if it does not break at some point. One of the most illustrative examples Such a graph is a sinusoid, which you can see in the picture in this section. A function is continuous at some point x0 if a number of conditions are met:
- a function is defined at a given point;
- the right and left limits at a point are equal;
- the limit is equal to the value of the function at point x0.
If at least one condition is not met, the function is said to fail. And the points at which the function breaks are usually called break points. An example of a function that will “break” when displayed graphically is: y=(x+4)/(x-3). Moreover, y does not exist at the point x = 3 (since it is impossible to divide by zero).
In the function that we are studying (y=1/3(x^3-14x^2+49x-36)) everything turned out to be simple, since the graph will be continuous.
Even, odd
Now examine the function for parity. First, a little theory. An even function is one that satisfies the condition f(-x)=f(x) for any value of the variable x (from the range of values). Examples include:
- module x (the graph looks like a daw, the bisector of the first and second quarters of the graph);
- x squared (parabola);
- cosine x (cosine).
Note that all of these graphs are symmetrical when viewed with respect to the y-axis (that is, the y-axis).
What then is called an odd function? These are those functions that satisfy the condition: f(-x)=-f(x) for any value of the variable x. Examples:
- hyperbola;
- cubic parabola;
- sinusoid;
- tangent and so on.
Please note that these functions are symmetrical about the point (0:0), that is, the origin. Based on what was said in this section of the article, even and odd function must have the property: x belongs to the set of definition and -x too.
Let's examine the function for parity. We can see that she doesn't fit any of the descriptions. Therefore, our function is neither even nor odd.
Asymptotes
Let's start with a definition. An asymptote is a curve that is as close as possible to the graph, that is, the distance from a certain point tends to zero. In total, there are three types of asymptotes:
- vertical, that is, parallel to the y-axis;
- horizontal, that is, parallel to the x axis;
- inclined.
As for the first type, these lines should be looked for at some points:
- gap;
- ends of the domain of definition.
In our case, the function is continuous, and the domain of definition is equal to R. Consequently, there are no vertical asymptotes.
The graph of a function has a horizontal asymptote, which corresponds to next requirement: if x tends to infinity or minus infinity, and the limit is equal to some number (for example, a). In this case, y=a is the horizontal asymptote. There are no horizontal asymptotes in the function we are studying.
An oblique asymptote exists only if two conditions are met:
- lim(f(x))/x=k;
- lim f(x)-kx=b.
Then it can be found using the formula: y=kx+b. Again, in our case there are no oblique asymptotes.
Function zeros
The next step is to examine the graph of the function for zeros. It is also very important to note that the task associated with finding the zeros of a function occurs not only when studying and constructing a graph of a function, but also as an independent task and as a way to solve inequalities. You may be required to find the zeros of a function on a graph or use mathematical notation.
Finding these values will help you graph the function more accurately. If we talk in simple language, then the zero of the function is the value of the variable x at which y = 0. If you are looking for the zeros of a function on a graph, then you should pay attention to the points at which the graph intersects with the x-axis.
To find the zeros of the function, you need to solve the following equation: y=1/3(x^3-14x^2+49x-36)=0. After carrying out the necessary calculations, we get the following answer:
Sign constancy
The next stage of research and construction of a function (graph) is finding intervals of constant sign. This means that we must determine at which intervals the function takes a positive value and at which intervals it takes a negative value. The zero functions found in the last section will help us do this. So, we need to build a straight line (separate from the graph) and in in the right order distribute the zeros of the function over it from smallest to largest. Now you need to determine which of the resulting intervals has a “+” sign and which has a “-”.
In our case, the function takes a positive value on intervals:
- from 1 to 4;
- from 9 to infinity.
Negative meaning:
- from minus infinity to 1;
- from 4 to 9.
This is quite easy to determine. Substitute any number from the interval into the function and see what sign the answer turns out to have (minus or plus).
Increasing and decreasing function
In order to explore and construct a function, we need to know where the graph will increase (go up along the Oy axis) and where it will fall (crawl down along the y-axis).
The function increases only if the larger value of the variable x corresponds to higher value u. That is, x2 is greater than x1, and f(x2) is greater than f(x1). And we observe a completely opposite phenomenon with a decreasing function (the more x, the less y). To determine the intervals of increase and decrease, you need to find the following:
- domain of definition (we already have);
- derivative (in our case: 1/3(3x^2-28x+49);
- solve the equation 1/3(3x^2-28x+49)=0.
After calculations we get the result:
We get: the function increases on the intervals from minus infinity to 7/3 and from 7 to infinity, and decreases on the interval from 7/3 to 7.
Extremes
The function under study y=1/3(x^3-14x^2+49x-36) is continuous and exists for any value of the variable x. The extremum point shows the maximum and minimum of a given function. In our case there are none, which greatly simplifies the construction task. Otherwise, they can also be found using the derivative function. Once found, do not forget to mark them on the chart.
Convexity and concavity
We continue to further explore the function y(x). Now we need to check it for convexity and concavity. The definitions of these concepts are quite difficult to comprehend; it is better to analyze everything using examples. For the test: a function is convex if it is a non-decreasing function. Agree, this is incomprehensible!
We need to find the derivative of a second order function. We get: y=1/3(6x-28). Now let's equate the right-hand side to zero and solve the equation. Answer: x=14/3. We found the inflection point, that is, the place where the graph changes from convexity to concavity or vice versa. On the interval from minus infinity to 14/3 the function is convex, and from 14/3 to plus infinity it is concave. It is also very important to note that the inflection point on the chart should be smooth and soft, no sharp corners should not be present.
Defining additional points
Our task is to investigate and construct a graph of the function. We have completed the study; constructing a graph of the function is now not difficult. For more accurate and detailed reproduction of a curve or straight line on the coordinate plane, you can find several auxiliary points. They are quite easy to calculate. For example, we take x=3, solve the resulting equation and find y=4. Or x=5, and y=-5 and so on. You can take as many additional points as you need for construction. At least 3-5 of them are found.
Plotting a graph
We needed to investigate the function (x^3-14x^2+49x-36)*1/3=y. All necessary marks during the calculations were made on the coordinate plane. All that remains to be done is to build a graph, that is, connect all the dots. Connecting the dots should be smooth and accurate, this is a matter of skill - a little practice and your schedule will be perfect.
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