How to graph the function y x m. IV

By plotting the points (0; 4), (1; 5), (-1; 5), (2; 8), (-2; 8) on coordinate plane and connecting them with a smooth curve, we get a parabola (colored line in Fig. 43). Please note - this is exactly the same parabola as y = x 2, but only shifted along the y axis by 4 scale units up. The vertex of the parabola is now at the point (0; 4), and not at the point (0; 0), as for the parabola y = x 2. The axis of symmetry is still the straight line x = 0, as was the case

parabolas y = x 2.

If we construct graphs of the functions y = x 2 and y = x 2 -2 in one coordinate system (Fig. 44), then we note that the second graph is obtained from the first by shifting (parallel translation) along the y axis by 2 scale units down.

The situation is exactly the same with the graphs of other functions. For example, the graph of the function y = 2x 2 - 3 is a parabola, which is obtained from the parabola y = 2x 2 by shifting (parallel translation) along the y axis by 3 scale units down (Fig. 45).

In general, the following statement is true: in order to construct a graph of the function y = f(x) + mт, where m is a given positive number, it is necessary to shift the graph of the function y = f(x) along the y-axis by m scale units up; to build a graph of the function y = f(x) - m, where m is a given positive number, you need to shift the graph of the function y = f(x) along the y axis by t units scale down.

By the way, this result is not completely new to you. Remember how things were with the graphs of the functions y = kx and y = kx + m: these are two parallel lines, one of which (y = kx) passes through the origin, and the other (y = kx + m) passes through the point ( 0; m), i.e., it was actually obtained from the first straight line by shifting along the y-axis by m units (Fig. 46).

Example 1. Draw a graph of the function y = -2x 2 + 5.

Solution. By constructing a parabola y = - 2x2 and moving it
along the y-axis upward by 5 units, we get graph of a function y = - 2x 2 + 5 (Fig. 47).

Example 2. Graph the function - 2
Solution. By constructing a hyperbola and shifting it downward along the y-axis by 2 units, we obtain a graph of the function - 2 (Fig. 48). Please note that both horizontal asymptote The hyperbola moved down 2 units: for the hyperbola the asymptote was the x axis (straight line y = 0), and for the hyperbola - 2 the asymptote was the straight line y = -2.

Example 3. Find the smallest and highest value functions y = 3 - 2x 2 on the interval [-2, 1].

Solution. Let's build a graph of the function y = 3 - 2x 2 and highlight its part on the segment [- 2, 1] (Fig. 49). We note that y nanom = - 5 (achieved at x = - 2),

and at the most = 3 (achieved at x = 0).

Answer: at our name. = - 5; at the most = 3.

1. The domain of definition of the function is the ray [-4, + oo).

2. y = 0 for x = - 2 and for x = 2; y > 0 at -4<х<-2 и при -2<х<2;у<0 при х > 2.

3. The function decreases on the intervals [-4, -2] and .
4. The function is limited above, but not limited below.

5- y nanom. does not exist; ymax = 4 (achieved at x = - 4 and at x = 0).

6. The function is continuous in a given domain of definition.

Comment. In fact, in this paragraph we were talking about plotting the function y = f (x) + m, where m is any number, both positive and negative. You probably noticed that when thinking about how many scale units we should shift the graph of the function y = f(x) along the y-axis, we did not pay attention to the sign of the number m; the graph actually shifted up or down by | m I units. But the direction of the shift was precisely determined by the sign of the number m at m > 0, the graph shifted upward, and at m< 0 - вниз.

Mordkovich A. G., Algebra. 8th grade: Textbook. for general education institutions. - 3rd ed., revised. - M.: Mnemosyne, 2001. - 223 p.: ill.

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In algebra lessons, students should learn how to build graphs. After all, these skills can be of good use in other subjects. This is also important for life situations. But for the material to be visual, different ways of presenting it are required. One of these effective ways is presentation.

slides 1-2 (Presentation topic “How to construct a graph of the function y=f(x)+m, if the graph of the function y=f(x) is known”, example)

This presentation was prepared by the author to facilitate the teacher’s work in order to save effort and time. There is everything necessary to convey the material to students. And this presentation begins with a special case, where an example shows how to build a graph of the function y=x2+4, if students can already build graphs of y=x2, a new graph is built based on it.

That is, you should first draw a graph of y=x2, and then, by constructing a table of values ​​for the second function, construct its graph. From this it becomes clear that the graph y=x2+4 is obtained by shifting the graph y=x2 up 4 units. Thus, the teacher can lead students to a way to construct such graphs, based on the skills acquired earlier. But this example only shows that a positive number is added to the standard function.

slides 3-4 (examples, construction rules)

But in mathematics you can often find a case where a negative number is added to a function. To do this, the author demonstrates two other examples located on the next slide. Moreover, the author focuses on the fact that the coefficient does not in any way affect the behavior of the graph in terms of its shift.

Further in the presentation there is a verbal formulation of the rule according to which the graph of the function y=f(x)+m is constructed. It says that if you add a positive number m to a function, then its graph shifts by m units up, and if you subtract it, then by m units down. Students should remember this well. Then there will be no problems with plotting graphs.

slides 5-6 (examples)

Next, using the example of a linear function passing through the origin, the author shows how to construct a graph if the number m is added to the value of the function. This will come in handy later, even in the same lesson.

After this, the author offers examples to show how to apply the acquired knowledge. Here the function y=-2x2+5 is given, you need to plot it. First, a graph y=-2x2 is constructed in the coordinate system, and then this graph, or rather, all its points rise 5 units up. This is all shown in the figure.

slides 7-8 (examples)

In the following example, you need to plot the function y=3/x-2. Here, too, the author suggests first constructing a graph y=3/x, and then lowering all points of this graph down 2 units, since the number 2 is subtracted from the value of the function.

But these examples are completely primitive, the level of which barely reaches a satisfactory level. In mathematics there are often tasks much more complex than these. Therefore, the author proposes to consider the following example, the essence of which is not to construct a graph of a function, but to find its largest and smallest values. But there is no way to do without the knowledge that was given in this presentation.

slides 9-10 (examples)

Then the work of finding the smallest and largest values ​​will be easier. First, you should construct a graph of this function according to the new rule, and then mark the interval in which you should find the smallest and largest values. And finally write down the answer.

The next example suggests solving an equation in which the left and right sides of the equality represent some functions. Therefore, to solve this equation you need to find the intersection points of these graphs. This means that graphs of functions need to be constructed. Just one of these functions is very similar to the one that fits the rule learned at the beginning of the lesson. By applying this rule, you will not have to build a table of values, which means the time required to complete this task will be reduced.

slides 11-12 (example, note)

The next example contains two rules for constructing function graphs. One of them students should go through the topics a little earlier, and the other according to this presentation. By constructing graphs according to these rules, more time remains to describe the properties of a given function. Which is also important for students.

And the last slide of the presentation contains the remark that the graph moves up or down by |m| units. The direction of the shift is determined by the sign of the number m: when m>0 the graph shifts upward, when m<0 - вниз.

The teacher, of course, can improve the presentation, but the author tried to place everything necessary and important here. It can be used in standard lessons, in demonstration lessons and in extracurricular activities. The material is selected harmoniously, according to the methodology of teaching mathematics.

Sections: Mathematics

Lesson objectives:

Educational:

• experimentally obtaining an algorithm for constructing graphs of functions of the form y = f (x+l), y = f (x)+m; ? y = k(x - l) 2 ,y = khx 2 +m, if the graph of the function is known y = kh 2 ;
• mastering basic concepts; influence of k, l and m on the movement of the graph;
• formation of generalized knowledge, methods of activity for constructing a graph of the function y = f (x+l), ? y = f (x)+m, if the graph of the function y= f(x) is known, operate with them;
• application of the resulting algorithm and the patterns of its manifestation to the construction of graphs of a quadratic function, a function of the form y= k/x.

Educational:

• improve the ability to think logically and express your thoughts out loud; stimulate the cognitive activity of students by setting a problem task, assessment and encouragement; promote the development of resourcefulness and intelligence;
• developing skills in working with graphs, identifying the properties of a function from a graph and using templates for constructing graphs;
• formation of the need to acquire knowledge, development of horizons, curiosity, attention.

Educational:

• to instill in students a desire to improve their knowledge; cultivate interest in the subject;
• developing self-control skills and habits of reflection;
• culture of mathematical speech, instilling interest in the study of mathematics.
• changing the role of the student in the educational process from a passive observer to an active researcher.

Equipment: ruler templates, pencil, colored pencils, tablets.

During the classes

Organizing time. /The goal is a positive attitude towards organized work/.

(Greeting, checking the student’s readiness for the lesson, organizing students’ attention).

Stage of checking homework completion. The goal is to identify gaps in knowledge.

Repetition of the topic: "How to construct a graph of the function y = f(x+l), if the graph of the function y = f(x) is known?" (Identification of typical deficiencies in knowledge and their causes; correction of mistakes; use of mutual assistance and self-control of students).

1. Name the formulas of the functions whose graphs you built in your homework, which differed from each other in the number added to the argument of the function. Name this number.

2. How are the graphs located in the first case relative to each other when plotted in the same coordinate system? By how many units is one of the graphs shifted relative to the other? (Shift the graph to the left).

3. Name the formulas of the functions whose graphs you built in your homework, which differed from each other by the number subtracted from the argument of the function. Name this number.

4. How are the graphs located in the second case relative to each other when plotted in the same coordinate system? By how many units is one of the graphs shifted relative to the other? (Graph shift to the right).

5. Given a graph of one of the functions, how could you graph the second function? What would you need to know to do this?

Updating knowledge. Oral work.

The goal is to distinguish between types of functions.

1. Task: Game. "Collect apples." Formulas are given on the inside of a paper model of an apple; you need to collect and sort the apples into baskets No. 1; No. 2; No. 3. In No. 1 - elementary functions; No. 2 - functions experiencing tension and compression; No. 3 - functions shifted along the OX axis. The graphs of which functions were in basket 1, 2 and 3? ( y = x 2 , y = (x+3) 2 ; y =4 (x+5) 2 ; y = (x+0.3) 2 ; y = (x - 3) 2 ; y = x 2 -2; y=2x 2 -3; y = -12x 2 y = - x 2 , , , ; y = - 2X 2 ; y = 0,5X 2). Slide 12. /Appendix No. 1/

Which formulas turned out to be superfluous? (Let’s put these formulas aside for now) / Appendix No. 2 /

2. Task. The graph of which function is shown in the figure.

3. Task. Fill in the empty columns of the table: /Appendix No. 3/

X	- 4	- 3	- 2	-	0	1			4
at		9		1			4	9

Based on the data obtained, what figure can be constructed? (curve).

What is the dependency formula? Construct its graph. What is the graph of a function called? y = kh 2? How are the branches of the parabola directed? For what argument values ​​does the function y = x 2 ?accepts: positive values; negative values?

The stage of preparing students for active and conscious learning of new material.

Masha and Misha (guys from the country "Young Mathematicians") argued among themselves. Masha said that to build a graph of the function y = f (x+l) you need to take many, many points and build a graph based on them, but Misha says that you can do it simpler. To do this, you can use a graph of a similar function and move it along the OX axis.

Guys, help them resolve this issue. Which guy is right and why? (Both are right, but the method of shifting along OX is more rational). Using the templates, use the tablet to construct graphs of the function y = (x+3) 2 ; y = (x-1) 2 ; y = - (x+3) 2. What conclusion can be drawn? How many points and which ones are more convenient to take to plot function graphs? (Work on 1 stencil, using templates - 1 part of the board). Slide13

Conclusion: "To build a graph of the function y = f(x+l), if the graph of the function y = f(x) is known: you need to shift the graph of the function y = f(x) - to the left if l>0? To the right if l<0".?(5точек, начальная противоположна l) l<0- вправо; l>0-left. /Appendix No. 4/

Masha and Misha looked at the function y = x 2 +2 and argued again: Masha claims that the graph of this function needs to be moved to the left along the OX axis, but Misha is firmly convinced that this cannot be done. Help Masha find the right solution. What do I need to do? (You need to plot a graph of the given function).

Learning new material. Reproduction of knowledge and methods of activity in new situations. (Constructed on the basis of the design method - 2.3 parts of the board). (Work in pairs).

So, the topic of our lesson: “How to construct a graph of the function y = f(x)+m, if the graph of the function y = f(x) is known. Let’s work in groups to accurately find out the algorithm for constructing this type of function. For each group, a similar with a given function the expression: 1gr - y = x 2 +2; 2nd gr. - y = x 2 - 2; 3rd gr. - y = - x 2 +2.

The construction and transformation of graphs of a quadratic function is underway. The guys will try to experimentally obtain an algorithm for plotting quadratic functions of the form y = khx 2 +m, using the graph of a function y = kh 2. (On the 2nd stencil - 3rd part of the board, the result of the solution is displayed using parabola templates)

(Goal: experimentally obtain an algorithm for constructing graphs of functions of the form: ? y = f (x)+m; ? y = k(x - l) 2 ,? y = khx 2 +m, if the graph of the function is known y = kh 2 ; establish the influence of k, l and m on the movement of the graph; learn methods of activity for constructing a graph of the function y = f (x+l), y = f (x)+m, functions of the form y= k/x, if the graph of the function y= f(x) is known, operate with them; then find out the properties of a given function from the graph, read and explain the graphs, see the beauty and perfection of the dependencies). Slide 22

Conclusion: "To build a graph of the function y = f(x) +m, if the graph of the function y = f(x) is known: you need to shift the graph of the function y = f(x) - up along the op-amp axis, if m>0? Down along axis of the op-amp, if m<0".? m>0 - up; m<0- вниз. / Приложение №5/

Let's check the output according to our friend - the textbook. Work according to the textbook. P.20 p.110. (The work itself is based on the textbook). Searching, reading and pronouncing an algorithm for working with functions of the form y = f(x+l), if the graph of the function y = f(x) is known. Repetition and application of the algorithm for working with linear functions. Analysis: example No. 1 and example No. 2. Page 112 textbooks

6.Analysis, presentation, evaluation and discussion of the results obtained. /The goal is the ability to see and explain in mathematical language the ongoing transformations/

But what about other types of functions, for example: the type y = 1/x - Masha doubts. Help Masha solve this problem. (See example No. 2, Page 112 of the textbook)

Let's apply the conclusion to solve different types of tasks.

"How to construct functions y=f(x+l) given the graph of the function y=f(x)?"

"How to construct functions y=f(x)+m, given the graph of the function y=f(x)?"

Solution: 1 gr. - No. 20.1(c); 1 gr.--20.1(d); 3 gr.-20.3 (c) according to our assistant - the problem book. Work according to the problem book. P.20 p.124.

Generalization of knowledge, its systematization. Reflection.

The goal is the ability to draw mathematical conclusions and explain their essence.

What general conclusion can be drawn for working with graphs y = f(x+l), y = f(x)+m, if the graph of the function y = f(x) is known. /Appendix No. 6/

Function type	The graph can be obtained from the graph of the function y=f(x) by shifting the original graph	The graph can be obtained from the graph of the function y=f(x) using a shift of the coordinate axis
y = f(x + l), l> 0	along the x-axis by l scale units to the left	ordinates by l scale units to the right
y = f(x + l), l< 0	along the x-axis by l scale units to the right	ordinates by l scale units to the left
y = f(x) + m, m> 0	along the ordinate axis m scale units up	abscissa down m scale units
y = f(x) + m, m< 0	along the ordinate axis m scale units down	abscissa up m scale units

Brilliant: if the answer is “When the graph is shifted to the right (left), in which direction does the ordinate axis shift?”, “When the graph is shifted down (up), in which direction does the abscissa shift?” Those. You can shift the coordinate axes for convenience.

Masha and Misha are satisfied with the output of the algorithm established by the 8th grade children, for a quadratic function and other types of functions of the form y = f(x+l), and y = f(x)+m, if the graph of the function y = f(x) is known.

8. Application of acquired knowledge. Checking your understanding of the essence of the topic being studied using educational test tests. Each student receives a worksheet on which they answer the question. At the end, students exchange sheets, check each other’s answers, and give a grade. For each correct task -1 point. Answers:/Appendix No. 7/

No. 1 - 1B, 2B, 3G, 4A. No. 2 - A4, B1, B2, G3. No. 3 - Ic, IIa, IIIb. No. 4 - Ib, IIa, IIIc. No. 5 - 13, 21, 34, 42.

Application of verification tests. /Appendix No. 8/

/Appendix No. 9/

Reflection. Summing up, homework (Goal - effectiveness of work in the lesson).

Reflective questions:

What new did you learn in class today?

What additional information would you like to know on the topic?

Did you like the lesson?

How do you evaluate your work in class?

Summing up, homework (Goal - effectiveness of work in the lesson).

• student work is analyzed;
• the best works are celebrated;
• students self-assessment of their activities is organized;
• the degree of compliance with the set goal and performance results is recorded;
• goals for subsequent activities are outlined (it is possible to create an equation of a function of the form y= f(x+l) +m, according to a given parabola graph) - on coordinate plane.