How to test a logarithmic function for monotonicity. Study of functions for monotonicity - Knowledge Hypermarket. Examples of studying a function for monotonicity

Lesson objectives:

Educational:

• repeat the description of the properties of a piecewise function according to the graph;
• derive and master formal definitions of increasing and decreasing functions;
• teach how to prove the monotonicity of a function on its domain of definition.

Educational:

• fostering cognitive interest;
• fostering a culture of communication;
• instilling responsibility for the common cause.

Educational:

• development of thinking and mathematical speech through the formulation of general conclusions and generalizations.

During the classes

Epigraph for the lesson:

“It’s not enough to have a good mind, the main thing is to use it well”
R. Descartes.

Homework for this lesson: find out which professions people have to read graphs based on their line of work.

Answers: - cardiologist (cardiogram)

Economist (graph of price growth dynamics, oil price growth, $ exchange rate growth)

Meteorologist (graph of temperature changes over the year)

Seismologist (graph of fluctuations in volcano activity, seismic activity in a given area).

Let's see how much we own this culture.

Auction "Chart Reading"

The last student to correctly name a function property gets a "5"

Additional auction:

A piece of the graph of which function is shown in the drawing?

Today in the lesson we will look in detail at only one property of a function - monotonicity.

Match the adjective “monotonous” with a noun. What do they mean by "monotonous"?

Movement.

Monotonous - which means? Same, repeating.

What property of a function can be associated with the phrase - monotonous movement? Movement where?

So: monotonicity is an increase and decrease of a function.

In the notebook: number, lesson topic "Study of a function for monotonicity."

Let's start with what we already know - the graph. Draw a coordinate system in each column and draw a graph of an arbitrary function that has the specified property over the entire domain of definition.

Table in the notebook:

Let's put the notebooks aside. To further study the property, let's make sure once again that we all understand well what we mean we're talking about at the lesson. We collect lotto.

Instructions: On each desk there is a table and a set of cards.

We work in pairs. There are more cards than necessary. Be careful. Collect the lotto on a notebook, so that when you turn it over, we read the coded phrase, the correctness of which depends on the well-coordinated work of each pair.

Set of cards:

After each pair adds up the lotto and turns the table over, the resulting words form the following phrase:

“From living contemplation to abstract thinking, from it to practice - this is the path to knowledge of truth” F. Engels.

On the side board:

Today we have to climb this ladder in order to comprehend only a small grain of the truth of the knowledge that humanity has accumulated on its path of development.

What stage do you think we are on? Contemplation, i.e. We look at the graphs. We continue working in the notebook, in the first column of the table.

Fix x 1, find the corresponding y 1 from the graph, fix x 2 - find y 2. Compare x 1 and x 2 (x 1< х 2). Что происходит со значением х?

Compare y 1 and y 2 (y 1 > y 2). What happens to the value of y?

Conclusion: A larger x value corresponds to a smaller y value. This is the definition of a decreasing function. Write it down in the table.

Independent work.

Option 1. Do the same operations in the second column of the table.

Option 2. Fill in the third column.

Check on the board and share results in pairs.

The result of the work.

If we know the definition, then we don’t need a graph to establish the type of monotonicity. This means that we have climbed to the second step on the ladder of knowledge.

All that remains is to apply your knowledge in practice.

V. Problem book p. 194, No. 4, 5. One student at the blackboard.

Given: y = 2x - 5

Prove: 1< у 2

Proof:

x 1< х 2 |· 2

2x 1< 2х 2 | + (- 5)

2x 1 - 5< 2х 2 - 5

at 1< у 2 >function y = 2x - 5 - increasing.

Given: y = 7 - 13x

Prove: y 1 > y 2

Proof: similar

What are the names of the functions we examined? What determines the type of monotonicity of a linear function? Record the output in a table. Using this conclusion, let’s perform No. 6 orally.

No. 8(a,b). According to the options, arrange them in a notebook according to the model.

Checking the output: what is the name of the function? What general formula defines the function? What determines the type of monotony? Write it down in the table.

Do you think the type of monotonicity will change if the graph is shifted along the Ox or Oy axis?

No. 8 (c, d) orally.

Remember the graphs of famous functions. Which of them behaves the same throughout the entire domain of definition? y = . Write it down in the table.

V. Our lesson is coming to an end. Close your notebooks. Open the diaries.

Homework:

on "3" - learn definitions 10., 32 No. 1,2;

to "4" + 32 No. 11.,

to "5" + task on the card.

Build graphs and you will get a picture. .

"dog"

x = 8, - 19 y - 3;

y = - x - 11, 0 x 8;

x = 0, - 19 y - 11;

y = - x - 19, - 14 x 0;

x = - 14, - 5 y 1;

y = - x -13, - 14 x - 8;

x = - 8, - 11 y - 5;

y = x - 3, - 8 x 0;

y = - 3.0 x 8;

y = - 0.6x + 1.2, - 2 x 8;

y = 1.7 x 10;

y = - 4x - 42.8, 8 x 10;

y = , 5 x 8;

y = - 0.4x + 8.0 x 2;

y = - 4x + 8.0 x 2.

"sailboat"

We first became acquainted with the concepts of increasing and decreasing functions in the 7th grade algebra course. Looking at the graph of the function, we took down the corresponding information: if, moving along the graph from left to right, we at the same time move from bottom to top (as if climbing a hill), then we declared the function to be increasing (Fig. 124); if we move from top to bottom (go down a hill), then we declared the function to be decreasing (Fig. 125).

However, mathematicians are not very fond of this method of studying the properties of a function. They believe that definitions of concepts should not be based on a drawing - the drawing should only illustrate one or another property of a function on its graph. Let us give strict definitions of the concepts of increasing and decreasing functions.

Definition 1. The function y = f(x) is said to be increasing on the interval X if, from the inequality x 1< х 2 - где хг и х2 — любые две точки промежутка X, следует неравенство f(x 1) < f(x 2).

Definition 2. The function y = f(x) is said to be decreasing on the interval X if the inequality x 1< х 2 , где х 1 и х 2 — любые две точки прс лежутка X, следует неравенство f(x 1) >f(x 2).
In practice, it is more convenient to use the following formulations:
the function increases if higher value the argument corresponds to a larger function value;
a function decreases if a larger value of the argument corresponds to a smaller value of the function.

Using these definitions and the properties of numerical inequalities established in § 33, we will be able to substantiate conclusions about the increase or decrease of previously studied functions.

1. Linear function y = kx +m

If k > O, then the function increases along the entire number line (Fig. 126); if k< 0, то функция убывает на всей числовой прямой (рис. 127).

Proof. Let f(x) = kx +m. If x 1< х 2 и k >Oh, then, according to the property of 3 numerical inequalities (see § 33), kx 1< kx 2 . Далее, согласно свойству 2, из kx 1 < kx 2
it follows that kx 1 + m< kx 2 + m, т. е. f(х 1) < f(х 2).

So, from the inequality x 1< х 2 следует, что f(х 1) < f(x 2). Это и означает возрастание функции у = f(х), т.е. linear function y = kx+ m.
If x 1< х 2 и k < 0, то, согласно свойству 3 числовых неравенств, kx 1 >kx 2 , and according to property 2, from kx 1 > kx 2 it follows that kx 1 + m> kx 2 + i.e.

So, from the inequality x 1< х 2 следует, что f(х 1) >f(x 2). This means a decrease in the function y = f(x), i.e., the linear function y = kx + m.

If a function increases (decreases) throughout its entire domain of definition, then it can be called increasing (decreasing) without indicating the interval. For example, about the function y = 2x - 3 we can say that it is increasing along the entire number line, but we can also say it more briefly: y = 2x - 3 - increasing
function.

2. Function y = x2

1. Consider the function y = x 2 on the ray. Let's take two non-positive numbers x 1 and x 2 such that x 1< х 2 . Тогда, согласно свойству 3 числовых
inequalities, the inequality - x 1 > - x 2 is satisfied. Since the numbers - x 1 and - x 2 are non-negative, then by squaring both sides of the last inequality, we obtain an inequality of the same meaning (-x 1) 2 > (-x 2) 2, i.e. This means that f(x 1) > f(x 2).
So, from the inequality x 1< х 2 следует, что f(х 1) >f(x 2).
Therefore, the function y = x 2 decreases on the ray (- 00, 0] (Fig. 128).

3. Function y

1. Consider a function on the interval (0, + 00).
Let x1< х 2 . Так как х 1 и х 2 — положительные числа, то из х 1 < x 2 следует (см. пример 1 из § 33), т. е. f(x 1) >f(x 2).
So, from the inequality x 1< х 2 следует, что f(x 1) >f(x 2). This means that the function decreases on the open ray (0, + 00) (Fig. 129).

2. Consider a function on the interval (-oo, 0). Let x 1< х 2 , х 1 и х 2 — отрицательные числа. Тогда - х 1 >- x 2, and both parts of the latter are unequal
properties are positive numbers, and therefore (we again used the inequality proven in Example 1 from § 33). Next we have, where we get from.
So, from the inequality x 1< х 2 следует, что f(x 1) >f(x 2) i.e. function decreases on the open ray (- 00 , 0)
Usually the terms “increasing function” and “decreasing function” are combined common name monotonic function, and the study of a function for increasing and decreasing is called the study of a function for monotonicity.

Solution.

1) Let’s plot the function y = 2x2 and take the branch of this parabola at x< 0 (рис. 130).

2) Let's build a graph of the function and highlight its part on the segment (Fig. 131).

3) Let's construct a hyperbola and select its part on the open ray (4, + 00) (Fig. 132).
4) Let us depict all three “pieces” in one coordinate system - this is the graph of the function y = f(x) (Fig. 133).
Let's read the graph of the function y = f(x).
1. The domain of definition of the function is the entire number line.

2. y = 0 at x = 0; y > 0 for x > 0.

3. The function decreases on the ray (-oo, 0], increases on the segment , decreases on the ray, is convex upward on the segment, convex downward on the ray (Fig. 128).

1. Consider a function on the interval (0, + 00).
Let x1< х 2 . Так как х 1 и х 2 - , то из х 1 < x 2 следует (см. пример 1 из § 33), т. е. f(x 1) >f(x 2).

So, from the inequality x 1< х 2 следует, что f(x 1) >f(x 2). This means that the function decreases on the open ray (0, + 00) (Fig. 129).

2. Consider a function on the interval (-oo, 0). Let x 1< х 2 , х 1 и х 2 - отрицательные числа. Тогда - х 1 >- x 2, and both sides of the last inequality are positive numbers, and therefore (we again used the inequality proven in example 1 from § 33). Next we have, where we get from.

So, from the inequality x 1< х 2 следует, что f(x 1) >f(x 2) i.e. function decreases on the open ray (- 00 , 0)

Usually the terms “increasing function” and “decreasing function” are combined under the general name monotonic function, and the study of a function for increasing and decreasing is called the study of a function for monotonicity.

Solution.

1) Let’s plot the function y = 2x2 and take the branch of this parabola at x< 0 (рис. 130).

2) Construct and select its part on the segment (Fig. 131).

3) Let's construct a hyperbola and select its part on the open ray (4, + 00) (Fig. 132).
4) Let us depict all three “pieces” in one coordinate system - this is the graph of the function y = f(x) (Fig. 133).

Let's read the graph of the function y = f(x).

1. The domain of definition of the function is the entire number line.

2. y = 0 at x = 0; y > 0 for x > 0.

3. The function decreases on the ray (-oo, 0], increases on the segment, decreases on the ray, is convex upward on the segment, convex downward on the ray)