Functions and graphs. How to build a parabola? What is a parabola? How are quadratic equations solved? Build function y ax2 bx c
Consider an expression of the form ax 2 + in + c, where a, b, c are real numbers, and is different from zero. it mathematical expression known as the square trinomial.
Recall that ax 2 is the leading term of this square trinomial, and is its leading coefficient.
But the square trinomial does not always have all three terms. Take for example the expression 3x 2 + 2x, where a=3, b=2, c=0.
Let's move on to quadratic function y \u003d ax 2 + in + c, where a, b, c are any arbitrary numbers. This function is quadratic because it contains a term of the second degree, that is, x squared.
It is quite easy to plot a quadratic function, for example, you can use the full square method.
Consider an example of plotting a function y equals -3x 2 - 6x + 1.
To do this, the first thing to remember is the scheme for highlighting the full square in the trinomial -3x 2 - 6x + 1.
We take out -3 from the first two terms in brackets. We have -3 times the sum of x plus 2x and add 1. Adding and subtracting the unit in brackets, we get the formula for the square of the sum, which can be collapsed. We get -3 times the sum (x + 1) squared minus 1, add 1. Expanding the brackets and adding like terms, the expression comes out: -3 times the square of the sum (x + 1) add 4.
Let's build a graph of the resulting function by going to the auxiliary coordinate system with the origin at the point with coordinates (-1; 4).
In the figure from the video, this system is indicated by dotted lines. We bind the function y equals -3x 2 to the constructed coordinate system. For convenience, we take control points. For example, (0;0), (1;-3), (-1;-3), (2;-12), (-2;-12). At the same time, we set aside them in the constructed coordinate system. The parabola obtained during the construction is the graph we need. In the figure, this is a red parabola.
Applying the full square selection method, we have a quadratic function of the form: y = a * (x + 1) 2 + m.
The graph of the parabola y \u003d ax 2 + bx + c is easy to obtain from the parabola y \u003d ax 2 by parallel translation. This is confirmed by a theorem that can be proved by taking the full square of the binomial. The expression ax 2 + bx + c after successive transformations turns into an expression of the form: a * (x + l) 2 + m. Let's draw a graph. Let's perform a parallel movement of the parabola y \u003d ax 2, combining the vertex with the point with coordinates (-l; m). The important thing is that x = -l, which means -b / 2a. So this line is the axis of the parabola ax 2 + bx + c, its vertex is at the point with the abscissa x, zero is equal to minus b divided by 2a, and the ordinate is calculated by the cumbersome formula 4ac - b 2 /. But it is not necessary to memorize this formula. Since, by substituting the value of the abscissa into the function, we get the ordinate.
To determine the axis equation, the direction of its branches and the coordinates of the parabola vertex, consider the following example.
Let's take the function y \u003d -3x 2 - 6x + 1. Having compiled the equation for the axis of the parabola, we have that x \u003d -1. And this value is the x-coordinate of the top of the parabola. It remains to find only the ordinate. Substituting the value -1 into the function, we get 4. The top of the parabola is at the point (-1; 4).
The graph of the function y \u003d -3x 2 - 6x + 1 was obtained by parallel transfer of the graph of the function y \u003d -3x 2, which means that it behaves similarly. The leading coefficient is negative, so the branches are directed downwards.
We see that for any function of the form y = ax 2 + bx + c, the easiest question is the last question, that is, the direction of the branches of the parabola. If the coefficient a is positive, then the branches are up, and if negative, then they are down.
The next most difficult question is the first question, because it requires additional calculations.
And the second one is the most difficult, because, in addition to calculations, knowledge of the formulas by which x is zero and y is zero is also needed.
Let's plot the function y \u003d 2x 2 - x + 1.
We determine immediately - the graph is a parabola, the branches are directed upwards, since the leading coefficient is 2, and this is a positive number. According to the formula, we find the abscissa x is zero, it is equal to 1.5. To find the ordinate, remember that zero is equal to a function of 1.5, when calculating we get -3.5.
Top - (1.5; -3.5). Axis - x=1.5. Take the points x=0 and x=3. y=1. Note these points. Based on three known points, we build the required graph.
To plot the function ax 2 + bx + c, you need:
Find the coordinates of the vertex of the parabola and mark them in the figure, then draw the axis of the parabola;
On the x-axis, take two points that are symmetrical about the axis of the parabola, find the value of the function at these points and mark them on the coordinate plane;
Through three points, construct a parabola, if necessary, you can take a few more points and build a graph based on them.
In the following example, we will learn how to find the largest and smallest values of the function -2x 2 + 8x - 5 on the segment.
According to the algorithm: a \u003d -2, b \u003d 8, then x zero is 2, and zero y is 3, (2; 3) is the top of the parabola, and x \u003d 2 is the axis.
Let's take the values x=0 and x=4 and find the ordinates of these points. This is -5. We build a parabola and determine that the smallest value of the function is -5 at x=0, and the largest is 3 at x=2.
The lesson on the topic “Function y=ax^2, its graph and properties” is studied in the 9th grade algebra course in the system of lessons on the topic “Functions”. This lesson requires careful preparation. Namely, such methods and means of training that will give truly good results.
The author of this video lesson took care to help teachers in preparing for lessons on this topic. He developed a video tutorial with all the requirements in mind. The material is selected according to the age of the students. It is not overloaded, but it is capacious enough. The author describes the material in detail, dwelling on more important points. Each theoretical point is accompanied by an example so that perception educational material was much more efficient and better quality.
The lesson can be used by a teacher in a regular algebra lesson in the 9th grade as a certain stage of the lesson - the explanation of new material. The teacher will not have to say or tell anything during this period. It is enough for him to turn on this video lesson and make sure that the students listen carefully and write down important points.
The lesson can also be used by students self-training to the lesson, as well as for self-education.
The duration of the lesson is 8:17 minutes. At the beginning of the lesson, the author notices that one of the important functions is the quadratic function. Then a quadratic function is introduced from a mathematical point of view. Its definition is given with explanations.
Further, the author introduces students to the domain of definition of a quadratic function. The correct math notation appears on the screen. After that, the author considers an example of a quadratic function in a real situation: a physical problem is taken as the basis, which shows how the path depends on time during uniformly accelerated motion.
After that, the author considers the function y=3x^2. The construction of the table of values of this function and the function y=x^2 appears on the screen. According to the data of these tables, graphs of functions are constructed. Here, an explanation appears in the box, how the graph of the function y=3x^2 from y=x^2 is obtained.
Having considered two special cases, an example of the function y=ax^2, the author comes to the rule of how the graph of this function is obtained from the graph y=x^2.
Next, we consider the function y=ax^2, where a<0. И, подобно тому, как строились графики функций до этого, автор предлагает построить график функции y=-1/3 x^2. При этом он строит таблицу значений, строит графики функций y=-1/3 x^2 и, замечая при этом закономерность расположения графиков между собой.
Then the consequences are deduced from the properties. There are four of them. Among them, a new concept appears - the vertices of a parabola. A remark follows, which says what transformations are possible for the graph of this function. After that, it is said how the graph of the function y=-f(x) is obtained from the graph of the function y=f(x), as well as y=af(x) from y=f(x).
This concludes the lesson containing the educational material. It remains to consolidate it by selecting the appropriate tasks depending on the abilities of the students.
Reference data on the exponential function are given - basic properties, graphs and formulas. The following questions are considered: domain of definition, set of values, monotonicity, inverse function, derivative, integral, power series expansion and representation by means of complex numbers.
ContentProperties of the exponential function
Exponential function y = a x , has the following properties on the set of real numbers () :
(1.1)
is defined and continuous, for , for all ;
(1.2)
when a ≠ 1
has many meanings;
(1.3)
strictly increases at , strictly decreases at ,
is constant at ;
(1.4)
at ;
at ;
(1.5)
;
(1.6)
;
(1.7)
;
(1.8)
;
(1.9)
;
(1.10)
;
(1.11)
,
.
Other useful formulas
.
The formula for converting to an exponential function with a different power base:
For b = e , we get the expression of the exponential function in terms of the exponent:
Private values
, , , , .
y = a x for different values of base a . The figure shows graphs of the exponential function
y (x) = x
for four values degree bases:a= 2
, a = 8
, a = 1/2
and a = 1/8
. It can be seen that for a > 1
exponential function is monotonically increasing. The larger the base of the degree a, the stronger the growth. At 0
< a < 1
exponential function is monotonically decreasing. The smaller the exponent a, the stronger the decrease.
Ascending, descending
The exponential function at is strictly monotonic, so it has no extrema. Its main properties are presented in the table.
| y = a x , a > 1 | y = x, 0 < a < 1 | |
| Domain | - ∞ < x < + ∞ | - ∞ < x < + ∞ |
| Range of values | 0 < y < + ∞ | 0 < y < + ∞ |
| Monotone | increases monotonically | decreases monotonically |
| Zeros, y= 0 | No | No |
| Points of intersection with the y-axis, x = 0 | y= 1 | y= 1 |
| + ∞ | 0 | |
| 0 | + ∞ |
Inverse function
The reciprocal of an exponential function with a base of degree a is the logarithm to base a.
If , then
.
If , then
.
Differentiation of the exponential function
To differentiate an exponential function, its base must be reduced to the number e, apply the table of derivatives and the rule for differentiating a complex function.
To do this, you need to use the property of logarithms
and the formula from the table of derivatives:
.
Let an exponential function be given:
.
We bring it to the base e:
We apply the rule of differentiation of a complex function. To do this, we introduce a variable
Then
From the table of derivatives we have (replace the variable x with z ):
.
Since is a constant, the derivative of z with respect to x is
.
According to the rule of differentiation of a complex function:
.
Derivative of exponential function
.
Derivative of the nth order:
.
Derivation of formulas > > >
An example of differentiating an exponential function
Find the derivative of a function
y= 35 x
Solution
We express the base of the exponential function in terms of the number e.
3 = e log 3
Then
.
We introduce a variable
.
Then
From the table of derivatives we find:
.
Because the 5ln 3 is a constant, then the derivative of z with respect to x is:
.
According to the rule of differentiation of a complex function, we have:
.
Answer
Integral
Expressions in terms of complex numbers
Consider the complex number function z:
f (z) = az
where z = x + iy ; i 2 = - 1
.
We express the complex constant a in terms of the modulus r and the argument φ :
a = r e i φ
Then
.
The argument φ is not uniquely defined. In general
φ = φ 0 + 2 pn,
where n is an integer. Therefore, the function f (z) is also ambiguous. Often considered its main importance
.
A bad teacher teaches the truth, a good teacher teaches to extract it.
A.Disterweg
Teacher: Netikova Margarita Anatolyevna, teacher of mathematics, school No. 471 of the Vyborgsky district of St. Petersburg.
Lesson topic: “Graph of a functiony= ax 2 »
Lesson type: learning lesson.
Target: teach students how to graph a function y= ax 2 .
Tasks:
Tutorials: develop the ability to build a parabola y= ax 2 and establish a pattern between the graph of the function y= ax 2
and coefficient a.
Developing: development of cognitive skills, analytical and comparative thinking, mathematical literacy, the ability to generalize and draw conclusions.
Educators: education of interest in the subject, accuracy, responsibility, exactingness to oneself and others.
Planned results:
Subject: be able to determine the direction of the branches of the parabola by the formula and build it using the table.
Personal: be able to defend their point of view and work in pairs, in a team.
Metasubject: be able to plan and evaluate the process and result of their activities, process information.
Pedagogical technologies: elements of problem-based and advanced learning.
Equipment: interactive whiteboard, computer, handouts.
1. The formula for the roots of a quadratic equation and the factorization of a quadratic trinomial.
2. Reduction of algebraic fractions.
3.Properties and function graph y= ax 2 , dependence of the direction of the branches of the parabola, its "expansion" and "compression" along the ordinate axis on the coefficient a.
Lesson structure.
1. Organizational part.
2.Updating knowledge:
Examination homework
Oral work according to ready-made drawings
3. Independent work
4.Explanation of new material
Preparation for learning new material (creating a problem situation)
Primary assimilation of new knowledge
5. Fixing
Application of knowledge and skills in a new situation.
6. Summing up the lesson.
7. Homework.
8. Lesson reflection.
Technological map of an algebra lesson in grade 9 on the topic: “Function graphy=
ax 2
»
| Lesson stages | Stage tasks | Teacher activity | Student activities | UUD |
| 1. Organizational part 1 minute | Creating a working mood at the beginning of the lesson | Greets with students checks their preparation for the lesson, notes those who are absent, writes the date on the board. | Preparing to work in the classroom, greet the teacher | Regulatory: organization of educational activities. |
| 2.Updating knowledge 4 minutes | Check homework, repeat and summarize the material studied in previous lessons and create conditions for the successful completion of independent work. | Collects notebooks from six students (selection of two from each row) to check homework for grade (Attachment 1), then works with the class on interactive whiteboard (appendix 2). | Six students hand in notebooks with homework for checking, then answer the questions of the frontal survey (appendix 2). | Cognitive: bringing knowledge into the system. Communicative: the ability to listen to the opinions of others. Regulatory: evaluation of the results of their activities. Personal: assessment of the level of assimilation of the material. |
| 3. Independent work 10 minutes | Check the ability to factorize a square trinomial, reduce algebraic fractions and describe some properties of functions according to its graph. | Gives students cards with an individual differentiated task (Appendix 3). and solution sheets. | Perform independent work, independently choosing the level of difficulty of exercises by points. | Cognitive: Personal: assessment of the level of assimilation of the material and their capabilities. |
| 4.Explanation of new material Preparing to learn new material Primary assimilation of new knowledge | Creating a favorable environment for getting out of a problem situation, perception and comprehension of new material, independent coming to the right conclusion | So, you know how to graph a function y= x 2 (charts are pre-built on three boards). Name the main properties of this function: 3. Vertex coordinates 5. Intervals of monotonicity What is the coefficient in this case? x 2 ? In the example of the square trinomial, you saw that this is not necessary at all. What sign can it be? Give examples. How parabolas with other coefficients will look like, you have to find out for yourself. The best way to study something is to discover for yourself. D.Poya We divide into three teams (in rows), choose the captains who go to the board. The task for the teams is written on three boards, the competition begins! In one coordinate system, construct graphs of functions 1 team: a) y=x 2 b) y= 2x 2 c) y= x 2 2 team: a) y \u003d - x 2 b) y \u003d -2x 2 c) y \u003d - x 2 3 team: a) y=x 2 b) y=4x 2 c) y=-x 2 Mission accomplished! (Annex 4). Find functions that have the same properties. Captains consult with their teams. What does it depend on? But how do these parabolas still differ and why? What determines the "thickness" of the parabola? What determines the direction of the branches of a parabola? We will conditionally call the schedule a) "initial". Imagine an elastic band: if you stretch it, it becomes thinner. This means that graph b) was obtained by stretching the original graph along the y-axis. How is graph c) obtained? So, at x 2 can be any coefficient that affects the configuration of the parabola. Here is the topic of our lesson: "Function Graphy= ax 2 » | 1. R 4. Branches up 5. Decreases by (- Increasing by )
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