Today we offer to research and build a function chart with us. After carefully studying this article, you don’t have to sweat for a long time to complete this kind of task. Investigating and plotting a function is not easy; the work is voluminous, requiring maximum attention and accuracy of calculations. To facilitate the perception of the material, we will gradually study the same function, explain all our actions and calculations. Welcome to the amazing and fascinating world of mathematics! Go!
Domain
In order to investigate and plot a function, you need to know several definitions. Function is one of the basic (basic) concepts in mathematics. It reflects the relationship 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 variation. So, y is a function of x, provided that each value of the second variable corresponds to one value of the second. Moreover, the variable y is dependent, and it is called a function. It is customary to say that the variables x and y are in a functional relationship. For greater clarity, this dependence is plotted function. What is a function graph? This is a set of points on the coordinate plane, where each value of x corresponds to one value of y. The graphs can be different - a straight line, a hyperbola, a parabola, a sinusoid and so on.
A function graph cannot be constructed without research. Today we will learn how to conduct research and plot a function. It is very important to make notes on the coordinate plane during the study. So to cope with the task will be much easier. The most convenient research plan:
- Domain.
- Continuity.
- Parity or oddness.
- Periodicity.
- Asymptotes.
- Zeros.
- Permanent sign.
- Ascending and Descending.
- Extremes.
- Bulge and concavity.
Let's start with the first paragraph. We find the domain of definition, that is, at 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 R. It can be written as follows x Î R.
Continuity
Now we will investigate the function of the gap. In mathematics, the term "continuity" appeared as a result of studying the laws of motion. What is infinite? Space, time, some dependencies (an example is the dependence of the variables S and t in motion problems), the temperature of a heated object (water, a frying pan, a thermometer, and so on), a continuous line (that is, one that can be drawn without taking it off the sheet pencil).
A continuous chart is one that does not break at some point. One of the most illustrative examples of such a graph is a sine wave, which you can see in the picture in this section. The function is continuous at some point x0 if a number of conditions are satisfied:
- at this point a function is defined;
- the right and left limits at the point are equal;
- the limit is equal to the value of the function at x0.
If at least one of the conditions is not met, they say that the function suffers a break. And the points at which the function breaks are commonly called break points. An example of a function that will “break” in a graphic display is: y = (x + 4) / (x-3). Moreover, y does not exist at the point x = 3 (since it is impossible to divide by zero).
Everything turned out to be simple in the function that we are investigating (y = 1/3 (x ^ 3-14x ^ 2 + 49x-36)), since the graph will be continuous.
Parity, oddness
Now examine the function for parity. First, a little theory. Even is called that function 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 is similar to a daw, the bisector of the first and second quarter of the graph);
- x squared (parabola);
- cosine x (cosine).
Note that all of these graphs are symmetrical if we consider this with respect to the ordinate axis (i.e., y).
And then what is called an odd function? Those are those functions that satisfy the condition: f (-x) = - f (x) for any value of the variable x. Examples:
- hyperbola;
- cubic parabola;
- sine wave;
- tangentoid and so on.
Please note that these functions are symmetric about the point (0: 0), that is, the origin. Based on what was said in this section of the article, an even and an odd function should have the property: x belongs to the set of definition and –x too.
We examine the function for parity. We may notice that it does not fit any of the descriptions. Therefore, our function is neither odd nor even.
Asymptotes
Let's start with the definition. Asymptot is a curve that is as close to the graph as possible, that is, the distance from a point tends to zero. There are three types of asymptotes in total:
- vertical, that is, parallel to the y axis;
- horizontal, that is, parallel to the x axis;
- inclined.
As for the first type, then these lines are worth looking at some points:
In our case, the function is continuous, and the domain of definition is R. Therefore, there are no vertical asymptotes.
The horizontal asymptote is the graph of the function, which meets the following requirement: if x tends to infinity or minus infinity, and the limit is equal to a certain number (for example, a). In this case, y = a - this is the horizontal asymptote. In the function we are studying, there are no horizontal asymptotes.
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 by the formula: y = kx + b. Again, in our case, there are no oblique asymptotes.
Zeros function
The next step is to examine the graph of the function by zeros. It is also important to note that the task associated with finding the zeros of a function is found not only in the study and construction of 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 the function on the graph or use a mathematical notation.
Finding these values will help you more accurately plot the function. In simple terms, the zero of a function is the value of the variable x, for which y = 0. If you are looking for zeros of a function in a graph, then you should pay attention to the points at which the graph intersects with the abscissa axis.
To find the zeros of a function, it is necessary to solve the following equation: y = 1/3 (x ^ 3-14x ^ 2 + 49x-36) = 0. After performing the necessary calculations, we get the following answer:
It is recommended to immediately mark the found points on the chart.
Constancy
The next stage in the study and construction of a function (graph) is to find intervals of constant sign. This means that we must determine at what intervals the function takes a positive value, and at which intervals it takes a negative value. This will help us to do the function zeros found in the last section. So, we need to build a straight line (separately from the graph) and distribute the zeros of the function from the smallest to the largest in it in the correct order. Now you need to determine which of the obtained intervals has a “+” sign, and which “-”.
In our case, the function takes a positive value at intervals:
- from 1 to 4;
- from 9 to infinity.
Negative meaning:
- from minus infinity to 1;
- from 4 to 9.
This is easy to determine. Substitute any number from the gap into the function and see with what sign the answer turned out (minus or plus).
Function increase and decrease
In order to investigate and construct a function, we need to find out where the graph will increase (go up the coordinate line Oy) and where it will fall (crawl down the ordinate).
The function increases only if a larger value of the variable x corresponds to a larger value of y. That is, x2 is greater than x1, and f (x2) is greater than f (x1). And we observe a completely opposite phenomenon in a decreasing function (the more x, the less y). To determine the intervals of increase and decrease, it is necessary to find the following:
- scope (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 in the intervals from the minus of infinity to 7/3 and from 7 to infinity, and decreases in the interval from 7/3 to 7.
Extremums
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 this function. In our case, there are none, which greatly simplifies the construction task. Otherwise, the extremum points are also found using the derivative function. After finding, do not forget to mark them on the chart.
Bulge and concavity
We continue to investigate the function y (x). Now we need to check it for bulge and concavity. The definitions of these concepts are hard enough to perceive, it is better to analyze everything with examples. For the test: the function is convex if it is an indefinite integral of a non-decreasing function. Agree, this is incomprehensible!
We need to find the derivative of the second-order function. We get: y = 1/3 (6x-28). Now we equate the right side to zero and solve the equation. Answer: x = 14/3. We found the inflection point, that is, the place where the graph changes the convexity to concavity or vice versa. In the interval from minus infinity to 14/3, the function is convex, and from 14/3 to plus infinity it is concave. It is very important to note that the inflection point on the graph should be smooth and soft, no sharp angles should be present.
Defining Additional Points
Our task is to research and plot the function. We finished the study, it’s easy to build a function graph now. For more accurate and detailed reproduction of a curve or line on the coordinate plane, you can find several auxiliary points. They are pretty easy to figure out. 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 to build. At least 3-5 of them are found.
Plotting
We needed to investigate the function (x ^ 3-14x ^ 2 + 49x-36) * 1/3 = y. All necessary marks during the calculations were applied on the coordinate plane. All that remains to be done is to build a chart, that is, to connect all the points together. Connecting the dots is smooth and accurate, this is a matter of skill - a little practice and your schedule will be perfect.