Charts piecewise given functions
Murzalieva T.A. teacher mathematicians MBOU"Borskaya average comprehensive school» Boksitogorsk district Leningrad region
Target:
- master the linear spline method for constructing graphs containing a module;
- learn to apply it in simple situations.
Under spline(from the English spline - plank, rail) is usually understood as a piecewise given function.
Such functions have been known to mathematicians for a long time, starting with Euler (1707-1783, Swiss, German and Russian mathematician), but their intensive study began, in fact, only in the middle of the 20th century.
In 1946, Isaac Schoenberg (1903-1990, Romanian and American mathematician) first time using this term. Since 1960 with development computer technology began using splines in computer graphics and modeling.
1 . Introduction
2. Definition of a linear spline
3. Module Definition
4. Graphing
One of the main purposes of functions is description real processes occurring in nature.
But for a long time, scientists - philosophers and natural scientists - have identified two types of processes: gradual ( continuous ) And spasmodic.
When a body falls to the ground, it first occurs continuous increase driving speed , and at the moment of collision with the surface of the earth speed changes abruptly , becoming equal to zero or changing the direction (sign) when the body “bounces” from the ground (for example, if the body is a ball).
But since there are discontinuous processes, then means of describing them are needed. For this purpose, functions are introduced that have ruptures .
a - by the formula y = h(x), and we will assume that each of the functions g(x) and h(x) is defined for all values of x and has no discontinuities. Then, if g(a) = h(a), then the function f(x) has a jump at x=a; if g(a) = h(a) = f(a), then the “combined” function f has no discontinuities. If both functions g and h are elementary, then f is called piecewise elementary. "width="640" - One way to introduce such discontinuities is next:
Let function y = f(x)
at x is defined by the formula y = g(x),
and when xa - formula y = h(x), and we will consider that each of the functions g(x) And h(x) is defined for all values of x and has no discontinuities.
Then , If g(a) = h(a), then the function f(x) has at x=a jump;
if g(a) = h(a) = f(a), then the "combined" function f has no breaks. If both functions g And h elementary, That f is called piecewise elementary.
Charts continuous functions
Graph the function:
Y = |X-1| + 1
X=1 – formula change point
Word "module" came from Latin word"modulus", which means "measure".
Modulus of numbers A called distance (in single segments) from the origin to point A ( A) .
This definition reveals geometric meaning module.
Module (absolute value ) real number A the same number is called A≥ 0, and opposite number -A, if a
0 or x=0 y = -3x -2 at x "width="640" Graph the function y = 3|x|-2.
By definition of the modulus, we have: 3x – 2 at x0 or x=0
-3x -2 at x
x n) "width="640" . Let x be given 1 X 2 X n – points of change of formulas in piecewise elementary functions.
A function f defined for all x is called piecewise linear if it is linear on each interval
and besides, the coordination conditions are met, that is, at the points of changing formulas, the function does not suffer a break.
Continuous piecewise linear function called linear spline . Her schedule There is polyline with two infinities extreme links – left (corresponding to the values x n ) and right ( corresponding values x x n )
A piecewise elementary function can be defined by more than two formulas
Schedule - broken line with two infinite extreme links - left (x1).
Y=|x| - |x – 1|
Formula change points: x=0 and x=1.
Y(0)=-1, y(1)=1.
It is convenient to plot the graph of a piecewise linear function, pointing on coordinate plane vertices of the broken line.
In addition to building n vertices should build Also two points : one to the left of the vertex A 1 ( x 1; y ( x 1)), the other - to the right of the top An ( xn ; y ( xn )).
Note that a discontinuous piecewise linear function cannot be represented as a linear combination of the moduli of binomials .
Graph the function y = x+ |x -2| - |X|.
A continuous piecewise linear function is called a linear spline
1.Points for changing formulas: X-2=0, X=2 ; X=0
2. Let's make a table:
U( 0 )= 0+|0-2|-|0|=0+2-0= 2 ;
y( 2 )=2+|2-2|-|2|=2+0-2= 0 ;
at (-1 )= -1+|-1-2| - |-1|= -1+3-1= 1 ;
y( 3 )=3+|3-2| - |3|=3+1-3= 1 .
Construct a graph of the function y = |x+1| +|x| – |x -2|.
1 .Points for changing formulas:
x+1=0, x=-1 ;
x=0 ; x-2=0, x=2.
2 . Let's make a table:
y(-2)=|-2+1|+|-2|-|-2-2|=1+2-4=-1;
y(-1)=|-1+1|+|-1|-|-1-2|=0+1-3=-2;
y(0)=1+0-2=-1;
y(2)=|2+1|+|2|-|2-2|=3+2-0=5;
y(3)=|3+1|+|3|-|3-2|=4+3-1=6.
|x – 1| = |x + 3|
Solve the equation:
Solution. Consider the function y = |x -1| - |x +3|
Let's build a graph of the function /using the linear spline method/
- Formula change points:
x -1 = 0, x = 1; x + 3 =0, x = - 3.
2. Let's make a table:
y(- 4) =|- 4–1| - |- 4+3| =|- 5| - | -1| = 5-1=4;
y( -3 )=|- 3-1| - |-3+3|=|-4| = 4;
y( 1 )=|1-1| - |1+3| = - 4 ;
y(-1) = 0.
y(2)=|2-1| - |2+3|=1 – 5 = - 4.
Answer: -1.
1. Construct graphs of piecewise linear functions using the linear spline method:
y = |x – 3| + |x|;
1). Formula change points:
2). Let's make a table:
2. Construct graphs of functions using the teaching aid “Live Mathematics” »
A) y = |2x – 4| + |x +1|
1) Formula change points:
2) y() =
B) Build function graphs, establish a pattern :
a) y = |x – 4| b) y = |x| +1
y = |x + 3| y = |x| - 3
y = |x – 3| y = |x| - 5
y = |x + 4| y = |x| + 4
Use the Point, Line, and Arrow tools on the toolbar.
1. “Charts” menu.
2. “Build a graph” tab.
.3. In the “Calculator” window, set the formula.
Graph the function:
1) Y = 2x + 4
1. Kozina M.E. Mathematics. 8-9 grades: collection elective courses. – Volgograd: Teacher, 2006.
2. Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova. Algebra: textbook. For 7th grade. general education institutions / ed. S. A. Telyakovsky. – 17th ed. – M.: Education, 2011
3. Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova. Algebra: textbook. For 8th grade. general education institutions / ed. S. A. Telyakovsky. – 17th ed. – M.: Education, 2011
4. Wikipedia, the free encyclopedia
http://ru.wikipedia.org/wiki/Spline
Piecewise functions are the functions specified different formulas on different numerical intervals. For example,
This notation means that the value of the function is calculated using the formula √x when x is greater than or equal to zero. When x is less than zero, the value of the function is determined by the formula –x 2. For example, if x = 4, then f(x) = 2, because in in this case the root extraction formula is used. If x = –4, then f(x) = –16, since in this case the formula –x 2 is used (first we square it, then we take into account the minus).
To plot a graph of such a piecewise function, first plot two different functions regardless of the value of x (i.e. on the entire number line of the argument). After this, only those parts that belong to the corresponding x ranges are taken from the resulting graphs. These parts of the graphs are combined into one. It is clear that in simple cases You can draw parts of the graphs at once, omitting the preliminary drawing of their “full” versions.
For the example above, for the formula y = √x, we get the following graph:
Here x in principle cannot accept negative values(i.e., the radical expression in this case cannot be negative). Therefore, the entire graph of the equation y = √x will go into the graph of the piecewise function.
Let's plot the function f(x) = –x 2 . We get an inverted parabola:
In this case, in the piecewise function we will take only that part of the parabola for which x belongs to the interval (–∞; 0). The result will be a graph of the piecewise function:
Let's look at another example:
The graph of the function f(x) = (0.6x – 0.5) 2 – 1.7 will be a modified parabola. The graph of f(x) = 0.5x + 1 is a straight line:
In a piecewise function, x can take values in a limited range: from 1 to 5 and from –5 to 0. Its graph will consist of two separate parts. We take one part on the interval from the parabola, the other on the interval [–5; 0] from straight line: 
Municipal budgetary educational institution
secondary school No. 13
"Piecewise functions"
Sapogova Valentina and
Donskaya Alexandra
Head Consultant:
Berdsk
1. Determination of main goals and objectives.
2. Questionnaire.
2.1. Determining the relevance of the work
2.2. Practical significance.
3. History of functions.
4. General characteristics.
5. Methods for specifying functions.
6. Construction algorithm.
8. Literature used.
1. Determination of main goals and objectives.
Target:
Find out a way to solve piecewise functions and, based on this, create an algorithm for their construction.
Tasks:
Get to know general concept about piecewise functions;
Find out the history of the term “function”;
Conduct a survey;
Identify ways to specify piecewise functions;
Create an algorithm for their construction;
2. Questionnaire.
A survey was conducted among high school students on their ability to construct piecewise functions. The total number of respondents was 54 people. Among them, 6% completed the work completely. 28% were able to complete the work, but with certain errors. 62% were unable to complete the work, although they made some attempts, and the remaining 4% did not start work at all.
From this survey we can conclude that the students of our school who are taking the program do not have a sufficient knowledge base, because this author does not pay attention to special attention for tasks of this kind. It is from this that the relevance and practical significance our work.
2.1. Determining the relevance of the work.
Relevance:
Piecewise functions are found both in the GIA and in the Unified State Exam; tasks that contain functions of this kind are scored 2 or more points. And, therefore, your assessment may depend on their decision.
2.2. Practical significance.
The result of our work will be an algorithm for solving piecewise functions, which will help to understand their construction. And it will increase your chances of getting the grade you want in the exam.
3. History of functions.
“Algebra 9th grade”, etc.;
Back forward
Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested this work, please download the full version.
Textbook: Algebra 8th grade, edited by A. G. Mordkovich.
Lesson type: Discovery of new knowledge.
Goals:
for the teacher goals are fixed at each stage of the lesson;
for the student:
Personal goals:
- Learn to clearly, accurately, competently express your thoughts verbally and writing, understand the meaning of the task;
- Learn to apply acquired knowledge and skills to solve new problems;
- Learn to control the process and results of your activities;
Meta-subject goals:
In cognitive activity:
- Development logical thinking and speech, the ability to logically substantiate one’s judgments and carry out simple systematizations;
- Learn to put forward hypotheses when problem solving, understand the need to check them;
- Apply knowledge in a standard situation, learn to perform tasks independently;
- Transfer knowledge to a changed situation, see the task in the context of the problem situation;
In information and communication activities:
- Learn to conduct a dialogue, recognize the right to a different opinion;
In reflective activity:
- Learn to anticipate possible consequences your actions;
- Learn to eliminate the causes of difficulties.
Subject goals:
- Find out what a piecewise function is;
- Learn to define a piecewise given function analytically from its graph;
During the classes
1. Self-determination educational activities
Purpose of the stage:
- include students in learning activities;
- determine the content of the lesson: we continue to repeat the topic of numerical functions.
Organization educational process at stage 1:
T: What did we do in previous lessons?
D: We repeated the topic of numerical functions.
U: Today we will continue to repeat the topic of previous lessons, and today we must find out what new things we can learn in this topic.
2. Updating knowledge and recording difficulties in activities
Purpose of the stage:
- update educational content, necessary and sufficient for the perception of new material: remember the formulas numerical functions, their properties and methods of construction;
- update mental operations, necessary and sufficient for the perception of new material: comparison, analysis, generalization;
- record an individual difficulty in an activity that demonstrates it personally significant level insufficiency of existing knowledge: specifying a piecewise given function analytically, as well as constructing its graph.
Organization of the educational process at stage 2:
T: The slide shows five numerical functions. Determine their type.
1) fractional-rational;
2) quadratic;
3) irrational;
4) function with module;
5) sedate.
T: Name the formulas corresponding to them.
3) 
;
4) 
;
U: Let's discuss what role each coefficient plays in these formulas?
D: Variables “l” and “m” are responsible for shifting the graphs of these functions left - right and up - down, respectively, the coefficient “k” in the first function determines the position of the branches of the hyperbola: k>0 - the branches are in the I and III quarters, k< 0 - во II и IV четвертях, а коэффициент “а” определяет направление ветвей параболы: а>0 - branches are directed upwards, and< 0 - вниз).
2. Slide 2
U: Define analytically the functions whose graphs are shown in the figures. (considering that they move y=x2). The teacher writes down the answers on the board.
D: 1) 
);
2)
;
3. Slide 3
U: Define analytically the functions whose graphs are shown in the figures. (considering that they are moving). The teacher writes down the answers on the board.
4. Slide 4
U: Using the previous results, define analytically the functions whose graphs are shown in the figures.
3. Identifying the causes of difficulties and setting goals for activities
Purpose of the stage:
- organize communicative interaction, during which the distinctive property a task that caused difficulty in learning activities;
- agree on the purpose and topic of the lesson.
Organization of the educational process at stage 3:
T: What is causing you difficulties?
D: Pieces of graphs are provided on the screen.
T: What is the purpose of our lesson?
D: Learn to define pieces of functions analytically.
T: Formulate the topic of the lesson. (Children try to formulate the topic independently. The teacher clarifies it. Topic: Piecewise defined function.)
4. Construction of a project for getting out of a difficulty
Purpose of the stage:
- organize communicative interaction to build a new mode of action, eliminating the cause of the identified difficulty;
- fix new way actions.
Organization of the educational process at stage 4:
T: Let's read the task carefully again. What results are asked to be used as help?
D: Previous ones, i.e. those written on the board.
U: Maybe these formulas are already the answer to this task?
D: No, because these formulas define the quadratic and rational function, and the slide shows their pieces.
U: Let's discuss what intervals of the x-axis correspond to the pieces of the first function?
U: Then analytical method the assignment of the first function looks like: if
T: What needs to be done to complete a similar task?
D: Write down the formula and determine which intervals of the abscissa axis correspond to the pieces of this function.
5. Primary consolidation in external speech
Purpose of the stage:
- record the studied educational content in external speech.
Organization of the educational process at stage 5:
7. Inclusion in the knowledge system and repetition
Purpose of the stage:
- train skills in using new content in conjunction with previously learned content.
Organization of the educational process at stage 7:
U: Define analytically the function whose graph is shown in the figure.
8. Reflection on activities in the lesson
Purpose of the stage:
- record new content learned in the lesson;
- evaluate your own activities in the lesson;
- thank your classmates who helped get the lesson results;
- record unresolved difficulties as directions for future educational activities;
- discuss and write down homework.
Organization of the educational process at stage 8:
T: What did we learn about in class today?
D: With a piecewise given function.
T: What work did we learn to do today?
D: Ask this type functions analytically.
T: Raise your hand, who understood the topic of today's lesson? (Discuss any problems that have arisen with the other children).
Homework
- No. 21.12(a, c);
- No. 21.13(a, c);
- №22.41;
- №22.44.
Real processes occurring in nature can be described using functions. Thus, we can distinguish two main types of processes that are opposite to each other - these are gradual or continuous And spasmodic(an example would be a ball falling and bouncing). But if there are discontinuous processes, then there are special means to describe them. For this purpose, functions are introduced that have discontinuities and jumps, that is, in different parts of the number line, the function behaves according to different laws and, accordingly, is specified by different formulas. The concepts of discontinuity points and removable discontinuity are introduced.
Surely you have already come across functions defined by several formulas, depending on the values of the argument, for example:
y = (x – 3, for x > -3;
(-(x – 3), at x< -3.
Such functions are called piecewise or piecewise specified. Sections of the number line with various formulas tasks, let's call them components domain. The union of all components is the domain of definition of the piecewise function. Those points that divide the domain of definition of a function into components are called boundary points. Formulas that define a piecewise function on each component of the domain of definition are called incoming functions. Charts piecewise defined functions are obtained as a result of combining parts of graphs constructed on each of the partition intervals.
Exercises.
Construct graphs of piecewise functions:
1)
(-3, at -4 ≤ x< 0,
f(x) = (0, for x = 0,
(1, at 0< x ≤ 5.
The graph of the first function is a straight line passing through the point y = -3. It originates at a point with coordinates (-4; -3), runs parallel to the x-axis to a point with coordinates (0; -3). The graph of the second function is a point with coordinates (0; 0). The third graph is similar to the first - it is a straight line passing through the point y = 1, but already in the area from 0 to 5 along the Ox axis.
Answer: Figure 1.
2)
(3 if x ≤ -4,
f(x) = (|x 2 – 4|x| + 3|, if -4< x ≤ 4,
(3 – (x – 4) 2 if x > 4.
Let's consider each function separately and build its graph.
So, f(x) = 3 is a straight line, parallel to the axis Oh, but you only need to depict it in the area where x ≤ -4.
Graph of the function f(x) = |x 2 – 4|x| + 3| can be obtained from the parabola y = x 2 – 4x + 3. Having constructed its graph, the part of the figure that lies above the Ox axis must be left unchanged, and the part that lies under the abscissa axis must be symmetrically displayed relative to the Ox axis. Then symmetrically display the part of the graph where
x ≥ 0 relative to the Oy axis for negative x. We leave the graph obtained as a result of all transformations only in the area from -4 to 4 along the abscissa axis.
The graph of the third function is a parabola, the branches of which are directed downward, and the vertex is at the point with coordinates (4; 3). We depict the drawing only in the area where x > 4.
Answer: Figure 2.
3)
(8 – (x + 6) 2, if x ≤ -6,
f(x) = (|x 2 – 6|x| + 8|, if -6 ≤ x< 5,
(3 if x ≥ 5.
The construction of the proposed piecewise given function is similar to previous point. Here the graphs of the first two functions are obtained from the transformations of the parabola, and the graph of the third is a straight line parallel to Ox.
Answer: Figure 3.
4) Graph the function y = x – |x| + (x – 1 – |x|/x) 2 .
Solution. The scope of this function is all real numbers, except zero. Let's expand the module. To do this, consider two cases:
1) For x > 0 we get y = x – x + (x – 1 – 1) 2 = (x – 2) 2.
2) At x< 0 получим y = x + x + (x – 1 + 1) 2 = 2x + x 2 .
Thus, we have a piecewise defined function:
y = ((x – 2) 2, for x > 0;
( x 2 + 2x, at x< 0.
The graphs of both functions are parabolas, the branches of which are directed upward.
Answer: Figure 4.
5) Draw a graph of the function y = (x + |x|/x – 1) 2.
Solution.
It is easy to see that the domain of the function is all real numbers except zero. After expanding the module, we obtain a piecewise given function:
1) For x > 0 we get y = (x + 1 – 1) 2 = x 2 .
2) At x< 0 получим y = (x – 1 – 1) 2 = (x – 2) 2 .
Let's rewrite it.
y = (x 2, for x > 0;
((x – 2) 2 , at x< 0.
The graphs of these functions are parabolas.
Answer: Figure 5.
6) Is there a function whose graph on the coordinate plane has common point from any straight line?
Solution.
Yes, it exists.
An example would be the function f(x) = x 3 . Indeed, the graph of a cubic parabola intersects with the vertical line x = a at point (a; a 3). Let now the straight line be given by the equation y = kx + b. Then the equation
x 3 – kx – b = 0 has real root x 0 (since a polynomial of odd degree always has at least one real root). Consequently, the graph of the function intersects with the straight line y = kx + b, for example, at the point (x 0; x 0 3).
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