Linear function. Line function Graph the function y x 2
Learn to take derivatives of functions. The derivative characterizes the rate of change of a function at a certain point lying on the graph of this function. In this case, the graph can be either a straight or curved line. That is, the derivative characterizes the rate of change of a function at a specific point in time. Remember the general rules by which derivatives are taken, and only then proceed to the next step.
- Read the article.
- How to take the simplest derivatives, for example, the derivative of an exponential equation, is described. The calculations presented in the following steps will be based on the methods described therein.
Learn to distinguish problems in which the slope coefficient needs to be calculated through the derivative of a function. Problems do not always ask you to find the slope or derivative of a function. For example, you may be asked to find the rate of change of a function at point A(x,y). You may also be asked to find the slope of the tangent at point A(x,y). In both cases it is necessary to take the derivative of the function.
Take the derivative of the function given to you. There is no need to build a graph here - you only need the equation of the function. In our example, take the derivative of the function f (x) = 2 x 2 + 6 x (\displaystyle f(x)=2x^(2)+6x). Take the derivative according to the methods outlined in the article mentioned above:
Substitute the coordinates of the point given to you into the found derivative to calculate the slope. The derivative of a function is equal to the slope at a certain point. In other words, f"(x) is the slope of the function at any point (x,f(x)). In our example:
If possible, check your answer on a graph. Remember that the slope cannot be calculated at every point. Differential calculus deals with complex functions and complex graphs where the slope cannot be calculated at every point, and in some cases the points do not lie on the graphs at all. If possible, use a graphing calculator to check that the slope of the function you are given is correct. Otherwise, draw a tangent to the graph at the point given to you and think about whether the slope value you found matches what you see on the graph.
- The tangent will have the same slope as the graph of the function at a certain point. To draw a tangent at a given point, move left/right on the X axis (in our example, 22 values to the right), and then up one on the Y axis. Mark the point, and then connect it to the point given to you. In our example, connect the points with coordinates (4,2) and (26,3).
The concept of a numerical function. Methods for specifying a function. Properties of functions.
A numeric function is a function that acts from one numeric space (set) to another numeric space (set).
Three main ways to define a function: analytical, tabular and graphical.
1. Analytical.
The method of specifying a function using a formula is called analytical. This method is the main one in the mat. analysis, but in practice it is not convenient.
2. Tabular method of specifying a function.
A function can be specified using a table containing the argument values and their corresponding function values.
3. Graphical method of specifying a function.
A function y=f(x) is said to be given graphically if its graph is constructed. This method of specifying a function makes it possible to determine the function values only approximately, since constructing a graph and finding the function values on it is associated with errors.
Properties of a function that must be taken into account when constructing its graph:
1) The domain of definition of the function.
Domain of the function, that is, those values that the argument x of the function F =y (x) can take.
2) Intervals of increasing and decreasing functions.
The function is called increasing on the interval under consideration, if a larger value of the argument corresponds to a larger value of the function y(x). This means that if two arbitrary arguments x 1 and x 2 are taken from the interval under consideration, and x 1 > x 2, then y(x 1) > y(x 2).
The function is called decreasing on the interval under consideration, if a larger value of the argument corresponds to a smaller value of the function y(x). This means that if two arbitrary arguments x 1 and x 2 are taken from the interval under consideration, and x 1< х 2 , то у(х 1) < у(х 2).
3) Function zeros.
The points at which the function F = y (x) intersects the abscissa axis (they are obtained by solving the equation y(x) = 0) are called zeros of the function.
4) Even and odd functions.
The function is called even, if for all argument values from the scope
y(-x) = y(x).
The graph of an even function is symmetrical about the ordinate.
The function is called odd, if for all values of the argument from the domain of definition
y(-x) = -y(x).
The graph of an even function is symmetrical about the origin.
Many functions are neither even nor odd.
5) Periodicity of the function.
The function is called periodic, if there is a number P such that for all values of the argument from the domain of definition
y(x + P) = y(x).
Linear function, its properties and graph.
A linear function is a function of the form y = kx + b, defined on the set of all real numbers.
k– slope (real number)
b– dummy term (real number)
x– independent variable.
· In the special case, if k = 0, we obtain a constant function y = b, the graph of which is a straight line parallel to the Ox axis passing through the point with coordinates (0; b).
· If b = 0, then we get the function y = kx, which is direct proportionality.
o The geometric meaning of the coefficient b is the length of the segment that the straight line cuts off along the Oy axis, counting from the origin.
o The geometric meaning of the coefficient k is the angle of inclination of the straight line to the positive direction of the Ox axis, calculated counterclockwise.
Properties of a linear function:
1) The domain of definition of a linear function is the entire real axis;
2) If k ≠ 0, then the range of values of the linear function is the entire real axis.
If k = 0, then the range of values of the linear function consists of the number b;
3) Evenness and oddness of a linear function depend on the values of the coefficients k and b.
a) b ≠ 0, k = 0, therefore, y = b – even;
b) b = 0, k ≠ 0, therefore y = kx – odd;
c) b ≠ 0, k ≠ 0, therefore y = kx + b is a function of general form;
d) b = 0, k = 0, therefore y = 0 is both an even and an odd function.
4) A linear function does not have the property of periodicity;
5) Points of intersection with coordinate axes:
Ox: y = kx + b = 0, x = -b/k, therefore (-b/k; 0) is the point of intersection with the x-axis.
Oy: y = 0k + b = b, therefore (0; b) is the point of intersection with the ordinate.
Comment. If b = 0 and k = 0, then the function y = 0 vanishes for any value of the variable x. If b ≠ 0 and k = 0, then the function y = b does not vanish for any value of the variable x.
6) The intervals of constant sign depend on the coefficient k.
a) k > 0; kx + b > 0, kx > -b, x > -b/k.
y = kx + b – positive at x from (-b/k; +∞),
y = kx + b – negative for x from (-∞; -b/k).
b)k< 0; kx + b < 0, kx < -b, x < -b/k.
y = kx + b – positive at x from (-∞; -b/k),
y = kx + b – negative for x of (-b/k; +∞).
c) k = 0, b > 0; y = kx + b is positive throughout the entire domain of definition,
k = 0, b< 0; y = kx + b отрицательна на всей области определения.
7) The monotonicity intervals of a linear function depend on the coefficient k.
k > 0, therefore y = kx + b increases throughout the entire domain of definition,
k< 0, следовательно y = kx + b убывает на всей области определения.
11. Function y = ax 2 + bx + c, its properties and graph.
| The function y = ax 2 + bx + c (a, b, c are constants, a ≠ 0) is called quadratic In the simplest case, y = ax 2 (b = c = 0) the graph is a curved line passing through the origin. The curve serving as a graph of the function y = ax 2 is a parabola. Every parabola has an axis of symmetry called the axis of the parabola. The point O of the intersection of a parabola with its axis is called the vertex of the parabola. |
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| The graph can be constructed according to the following scheme: 1) Find the coordinates of the vertex of the parabola x 0 = -b/2a; y 0 = y(x 0). 2) We construct several more points that belong to the parabola; when constructing, we can use the symmetries of the parabola relative to the straight line x = -b/2a. 3) Connect the indicated points with a smooth line. Example. Graph the function b = x 2 + 2x - 3. Solutions. The graph of the function is a parabola, the branches of which are directed upward. The abscissa of the vertex of the parabola x 0 = 2/(2 ∙1) = -1, its ordinates y(-1) = (1) 2 + 2(-1) - 3 = -4. So, the vertex of the parabola is point (-1; -4). Let's compile a table of values for several points that are located to the right of the axis of symmetry of the parabola - straight line x = -1. Function properties.
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Lesson topic: Function y =k x 2 , its properties and graph .
The purpose of the lesson: generalize and systematize knowledge about the quadratic function, its properties and graph
Educational objectives:
consolidate the basic properties of the quadratic function y =kx 2 and its graph using computer modeling and an interactive whiteboard.
solving mathematical problems using several methods and methods, identifying the advantages and disadvantages of each of them.
Developmental tasks
development of students' communication abilities,
development of intellectual and research culture of students,
development of skills in computer modeling and working on an interactive whiteboard
Educational tasks:
develop respect for other people's opinions
serious and responsible attitude to educational work.
Lesson type: lesson presentation, workshop.
Teaching methods: conversation, explanation, business game, demonstration, computer simulation, practical work.
Forms of organizing work with students: individual, frontal, pair (group).
Equipment: computer, multimedia projector, interactive whiteboard, regular board, graph paper, handouts: multi-level tasks, a memo with the requirements for performing practical work.
Software: presentation prepared V Microsoft PowerPoint; Advanced Grapher 1.62 (Multifunctional program for studying mathematical functions with a convenient graphical interface. Allows you to build graphs of functions and their derivatives, find extrema of functions and roots of equations, carry out integration, obtain a table of function values according to its formula, etc., status: freeware, copyright: SerpikSoft, website: ); interactive whiteboard software.
Lesson plan.
1. Organizational moment – 1-2 minutes.
2. Setting goals and objectives for the lesson – 2 min.
3. Equipment – 1 min.
4. Repetition of previously studied material – 10 min.
task No. 1
task No. 2
5. Practical work – 25 min.
Task No. 3
Defense of completed task No. 3
Task No. 4
Defense of completed task No. 4
6. Homework – 2 min.
7. Summing up the lesson. Grading – 3 min.
During the classes
Slide 1 is shown.
Stage I. Organizing time.
The teacher greets the children, notes those who are absent, checks the availability of drawing tools, handouts: task cards, graph paper, reminders.
Setting the goal and objectives of the lesson
Shown slide 2-5
Teacher. Today we will summarize and test the acquired knowledge and skills in practice, expand and systematize knowledge about the quadratic function y = kx 2 , as one of the mathematical models. Let's continue to master the capabilities of the interactive whiteboard, using a computer in our work, and consider constructing graphs of quadratic functions using it.
In real life, there are processes described by various mathematical models of the form y = f ( x ), G de f ( x ) - function. In the 7th grade we became acquainted with the linear function, in the 8th grade we began to get acquainted with another mathematical model, having studied f ( x ) – quadratic function. Let's check how you learned to distinguish one model from another in the first task.
Stage II. Repetition.
Task 1. Label the graph of the function.
For each graph shown on the interactive whiteboard, find the corresponding function.
Slide 6 shown
On the interactive whiteboard, students along the chain, using the method of moving objects (names of functions) from the gallery of drawings, move the functions to the corresponding graph, while justifying their choice.
The remaining students in a notebook and two on a regular board simultaneously write functions in two columns of the table, indicating the corresponding value k And b . The work is summarized. Students carry out mutual testing (on the interactive and regular boards, in notebooks).
Classification by type of mathematical model
y = kx + b
y = kx 2
y = 3x + 2 ; k = 3 b = 2
y =3x 2 ; k = 3
y =2x ; k =2 b =0
y = - 3x 2 ; k =-3
y =2x ; k =2 b =0
y = x 2 ; k =1
straight
parabola
Task 2. List the properties of a quadratic function.
Slide 7 shown
Teacher. In mathematics, it is important to distinguish one model from another, knowing the properties of each, and being able to use different languages (verbal, symbolic, graphic) when describing these properties. In preparation for the lesson, a group of children systematized general information about the quadratic function into a table using symbolic language. On the interactive whiteboard, the function properties table is covered with a curtain. Let's remember what we know about the properties of the quadratic function.
After a frontal survey to list the properties of a quadratic function, using the curtain technique from left to right, the first column of the table opens. The guys check the table to see if all the properties have been named. Then the properties of the function are listed depending on the coefficient; during the conversation, the rows of the table are simultaneously opened - the technique of moving the curtain down.
The students' answers are heard and the results of the repetition of the properties of the quadratic function are summarized. Students exercise self-control.
Stage III. Application of knowledge and skills
Practical work
Slide 8 shown
Task No. 3. “Construct and describe the properties of a piecewise given function
Teacher. So, now we will try to put all the knowledge into practice in different ways.
You will now be divided into three groups:
Group No. 1 “programmers”» – build a graph of a function using a computer.
Group No. 2 “practices”– build a graph of a function without using a computer on graph paper.
Group No. 3 “theorists” – describe the properties of a given function.
For the children of group No. 1 (attending an elective course in IVT), a work algorithm for computer modeling is displayed on the interactive board ( Slide 9 is shown), Group No. 2 uses the memo slide 23, application No. 2) , Group No. 3 has on the table a ready-made graph of this function, completed in advance by students in the IVT elective ( slide 14 ).
The task for children in group No. 2, with below average abilities, is divided into subtasks. Weak students build a graph of only one quadratic function, stronger students build a graph of a quadratic and a linear function, advanced students complete the entire task in its entirety.
The teacher checks the assignment for the students who completed the assignment first in each group. Then, as the practical work is completed, students check each other’s assignments in a chain. This way, all students' work will be checked. Those students who are experiencing difficulties turn to the teacher or the comrades of the neighboring pair for help.
Slide 10-15 is shown
Protection of completed work
Each group identifies a leader responsible for protecting the work. Students analyze the stages of constructing and describing the properties of a function. Students of group No. 2 exercise self-control by comparing their graph with the graph on the interactive whiteboard, constructed using computer modeling by students of group No. 1. Students of group No. 3 comment on the properties of the function, the graph of which is presented on the board.
During the defense, the teacher asks questions that help identify the advantages and disadvantages of each method of graphing a function:
What is the advantage of this method of graphing a function?
What disadvantages of this method can you name?
Protecting work done using a computer
Slide 16 shown
Advantages of the method:
Visualization, speed of work, accuracy of construction, ease of implementation, the ability to automate verification of the result; a schedule is created not only on paper, but also in electronic form.
Disadvantages of this method:
Computational skills are not being improved, there is no connection with theory, there is no availability of hardware and software.
Slide 17 shown
Protecting work done without a computer
Advantages of the method:
Independence from computer technology when used; development of computational skills, connection with theory.
Disadvantages of this method:
The work takes a long time, there is no precision in the construction, it is impossible to automate the verification of the result; The chart is created only on paper.
Task No. 4 "Solve the equationx 2 = 4 x - 4"
Slide 18 shown
Teacher. We invite you to solve the equation using two methods: graphical and analytical.
1. Graphic method - in two ways (computer modeling and without the help of a computer).
2. Method – analytical.
By analyzing the stages of graphically solving an equation, students formulate an algorithm for completing the task. Slide 19 shown
When using the analytical solution method, it is necessary to remember the formula for the square of the difference of two expressions.
The graphical solution method can be presented in two ways using computer modeling and traditionally.
The task is performed by students of groups No. 1-3 according to the same scheme as when performing practical work of task No. 3. Students complete the task and compare the result.
Protection of completed work.
A group of guys working at a computer demonstrate the result of their work using a multimedia projector on an interactive whiteboard, indicating the point of intersection of the function graphs and signing its coordinates. Group of students No. 3 - “theorists”, the decision is made on a regular board. Group of students No. 3 – “practitioners”, check the results with the interactive board.
Slide 20 shown
Teacher gives a task compare the results. Determine in your opinion a more effective method.
Stage IV. Homework.
Slide 21 shown
Teacher. In class you worked in groups, in pairs, doing one task together. At home you will have to do practical work based on your abilities. The task is differentiated by difficulty levels ( slide 22 - Appendix 2, slide 23 ). A slide with instructions for completing the work is shown on the board.
Stage V. Summing up the lesson. Grading.
Slide 24 shown
Today we have summarized and systematized knowledge on the topic “Function y = x 2, its properties and graph” using computer modeling and an interactive whiteboard, examined the solution of a mathematical problem in several ways, and found out the advantages and disadvantages of each method. For you, a more universal method turned out to be the use of mathematical modeling. However, the choice of a specific method also depends on the goals that we set when solving a particular problem. Different mathematical problems give us the opportunity to apply different techniques, methods and methods for specific practical problems. And you have the right to choose those that will be more suitable under the given conditions. In the next lesson, we move on to getting acquainted with a new mathematical model, replenishing the stock of functions being studied. All the knowledge and skills gained from constructing function graphs in two ways will help you in your future work. Thanks everyone for your work.
Literature
Magazine "Mathematics at School", No. 10, 2008
Journal "Informatics and Education", No. 10, 2008.
A.G. Mordkovich. Algebra 8th grade. Part 1. Textbook. M.: Mnemosyne, 2005.
A.G. Mordkovich. Algebra 8th grade. Part 2. Problem book. M.: Mnemosyne, 2005.
L.A.Alexandrova. Algebra 8th grade. Independent works / ed. A.G. Mordkovich. M.: Mnemosyne, 2006.
A.G. Mordkovich. Algebra 7-9. Methodological manual for teachers. M.: Mnemosyne, 2000.
Annex 1
Memo
1. How to graph a function.
Create a table of values.
Construct points on the coordinate plane.
Connect the points with a smooth line.
Label the graph of the function.
2. How to find the value of a function f (x ) on schedule.
Find the corresponding value of the variable on the x-axis.
Draw a perpendicular to the graph of the function and fix a point on it.
From this point, draw a perpendicular to the ordinate axis.
Axis intersection point at – and is the value of the function f ( x ).
3. How to check whether a point belongs to the graph of a function.
Find the value of the function from the abscissa of the point.
Compare the result with the ordinate of the point.
If the values coincide, the point belongs to the graph of the function.
Appendix 2
Practical work
Option A
1. Graph the function y = 2 X 2
a) meaning at at x = -1; 2; 1/2
b) value X , if y = -8
V) y max. And y name on the segment [-1; 2]
3. Does point A (-5; 50) belong to the graph of the function?
Option B
1. Graph the function y = - 0.5 X 2
2. For this function, find:
a) meaning at at x = -2; 0; 3
b) value X if y = - 8
V) y max. And y name on the segment [- 4; 0]
3. Does point A belong to the graph of the function (-10; - 50)
Option C
1. Graph the function y = 3/2 X 2
2. For this function, find:
a) meaning at at x = 2; 1; 2/ 3
b) value X if y = 6
V) y max. And y name on the segment [- 2; 1]
3. Does point A (-8;- 96) belong to the graph of the function?
Definition of a Linear Function
Let us introduce the definition of a linear function
Definition
A function of the form $y=kx+b$, where $k$ is nonzero, is called a linear function.
The graph of a linear function is a straight line. The number $k$ is called the slope of the line.
When $b=0$ the linear function is called a function of direct proportionality $y=kx$.
Consider Figure 1.
Rice. 1. Geometric meaning of the slope of a line
Consider triangle ABC. We see that $ВС=kx_0+b$. Let's find the point of intersection of the line $y=kx+b$ with the axis $Ox$:
\ \
So $AC=x_0+\frac(b)(k)$. Let's find the ratio of these sides:
\[\frac(BC)(AC)=\frac(kx_0+b)(x_0+\frac(b)(k))=\frac(k(kx_0+b))((kx)_0+b)=k \]
On the other hand, $\frac(BC)(AC)=tg\angle A$.
Thus, we can draw the following conclusion:
Conclusion
Geometric meaning of the coefficient $k$. The angular coefficient of the straight line $k$ is equal to the tangent of the angle of inclination of this straight line to the $Ox$ axis.
Study of the linear function $f\left(x\right)=kx+b$ and its graph
First, consider the function $f\left(x\right)=kx+b$, where $k > 0$.
- $f"\left(x\right)=(\left(kx+b\right))"=k>0$. Consequently, this function increases over the entire domain of definition. There are no extreme points.
- $(\mathop(lim)_(x\to -\infty ) kx\ )=-\infty $, $(\mathop(lim)_(x\to +\infty ) kx\ )=+\infty $
- Graph (Fig. 2).
Rice. 2. Graphs of the function $y=kx+b$, for $k > 0$.
Now consider the function $f\left(x\right)=kx$, where $k
- The domain of definition is all numbers.
- The range of values is all numbers.
- $f\left(-x\right)=-kx+b$. The function is neither even nor odd.
- For $x=0,f\left(0\right)=b$. When $y=0.0=kx+b,\ x=-\frac(b)(k)$.
Intersection points with coordinate axes: $\left(-\frac(b)(k),0\right)$ and $\left(0,\ b\right)$
- $f"\left(x\right)=(\left(kx\right))"=k
- $f^("")\left(x\right)=k"=0$. Therefore, the function has no inflection points.
- $(\mathop(lim)_(x\to -\infty ) kx\ )=+\infty $, $(\mathop(lim)_(x\to +\infty ) kx\ )=-\infty $
- Graph (Fig. 3).
The linear function y = kx + m when m = 0 takes the form y = kx. In this case, you can notice that:
- If x = 0, then y = 0. Therefore, the graph of the linear function y = kx passes through the origin, regardless of the value of k.
- If x = 1 then y = k.
Let's consider different values of k, and how y changes from this.
If k is positive (k > 0), then the straight line (the graph of the function), passing through the origin, will lie in the I and III coordinate quarters. After all, with positive k, when x is positive, then y will also be positive. And when x is negative, y will also be negative. For example, for the function y = 2x, if x = 0.5, then y = 1; if x = –0.5, then y = –1.
Now, assuming k is positive, consider three different linear equations. Let these be: y = 0.5x and y = 2x and y = 3x. How does the value of y change for the same x? Obviously it increases with k: the larger k, the larger y. This means that the straight line (function graph) with a larger value of k will have a larger angle between the x-axis (abscissa axis) and the function graph. Thus, the angle at which the straight axis intersects the x axis depends on k, and hence k is spoken of as slope of linear function.
Now let's study the situation when k x is positive, then y will be negative; and vice versa: if x y > 0. Thus, the graph of the function y = kx for at k
Suppose there are linear equations y = –0.5x, y = –2x, y = –3x. For x = 1 we get y = –0.5, y = –2, y = –3. For x = 2 we get y = –1, y = –2, y = –6. Thus, the larger k, the larger y if x is positive.
However, if x = –1, then y = 0.5, y = 2, y = 3. For x = –2 we get y = 1, y = 4, y = 6. Here, as the value of k decreases, y at x increases
Graph of the function at k
Graphs of functions of the type y = kx + m differ from graphs y = km only in a parallel shift.
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