Many tasks need to calculate the maximum or minimum quadratic function. Maximum or minimum can be found if the source function is recorded in a standard form: or through the coordinates of the vertex parabola: f (x) \u003d a (x - h) 2 + k (\\ displaystyle f (x) \u003d a (x - h) ^ (2) + K). Moreover, the maximum or minimum of any quadratic function can be calculated using mathematical operations.
Steps
The quadratic function is recorded in standard form
- In the function of the coefficients a \u003d 1 (\\ displaystyle a \u003d 1) and B \u003d 10 (\\ DisplayStyle B \u003d 10)
- As a second example, consider the function. Here a \u003d - 3 (\\ displaystyle a \u003d -3) and B \u003d 6 (\\ DisplayStyle B \u003d 6). Therefore, the "X" coordinate of the top of parabolas will calculate this:
-
Find the corresponding value F (X). Submold the found value "x" into the source function to find the corresponding value f (x). So you will find a minimum or maximum function.
- In the first example f (x) \u003d x 2 + 10 x - 1 (\\ displaystyle f (x) \u003d x ^ (2) + 10x-1) You have calculated that the coordinate "X" of the pearable parabol is equal to x \u003d - 5 (\\ displayStyle x \u003d -5). In the original function instead X (\\ DisplayStyle X) Put - 5 (\\ DisplayStyle -5)
- In the second example f (x) \u003d - 3 x 2 + 6 x - 4 (\\ displaystyle f (x) \u003d - 3x ^ (2) + 6x-4) You found that the coordinate "x" of the top of the parabola is equal to x \u003d 1 (\\ displaystyle x \u003d 1). In the original function instead X (\\ DisplayStyle X) Put 1 (\\ DISPLAYSTYLE 1)To find its maximum value:
-
Write down the answer. Re-read the condition of the task. If you need to find the coordinates of the top of the parabola, in response, write both values X (\\ DisplayStyle X) and Y (\\ DisplayStyle Y) (or f (x) (\\ DisplayStyle F (X))). If you need to calculate a maximum or minimum function, in response, write down only the value Y (\\ DisplayStyle Y) (or f (x) (\\ DisplayStyle F (X))). Once again, look at the sign of the coefficient A (\\ DisplayStyle a)To verify that you have calculated: a maximum or a minimum.
The quadratic function is recorded through the coordinates of the vertex parabola
-
Record the quadratic function through the coordinates of the parabola vertex. Such an equation is as follows:
Determine the direction of parabola. To do this, look at the coefficient sign A (\\ DisplayStyle a). If the coefficient A (\\ DisplayStyle a) Positive, parabola is directed up. If the coefficient A (\\ DisplayStyle a) Negative, parabola is directed down. For example:
Find the minimum or maximum function value. If the function is recorded through the coordinates of the pearabela vertex, minimum or maximum equal to the value of the coefficient K (\\ DisplayStyle K). In the examples above:
Find the coordinates of the pearabela vertices. If the task is required to find the top of the parabola, its coordinates are equal (H, K) (\\ DisplayStyle (H, K)). Note when the quadratic function is recorded through the coordinates of the pearabela vertex, the subtraction operation must be enclosed in brackets. (X - H) (\\ DisplayStyle (X - H)), Therefore, the value H (\\ DisplayStyle H) Takes up with the opposite sign.
How to calculate a minimum or maximum with the help of mathematical operations
First consider the standard type of equation. Record the quadratic function in the standard form: f (x) \u003d a x 2 + b x + c (\\ displaystyle f (x) \u003d ax ^ (2) + bx + c). If necessary, bring similar members and rearrange them to obtain a standard equation.
Find the first derivative. The first derivative of the quadratic function, which is recorded in a standard form, is equal to f '(x) \u003d 2 a x + b (\\ displaystyle f ^ (\\ prime) (x) \u003d 2AX + B).
Derivative equate to zero. Recall that the derived function is equal to the angular coefficient of the function at a certain point. In a minimum or maximum, the angular coefficient is zero. Therefore, to find the minimum or maximum function value, the derivative must be equal to zero. In our example:
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Write down the function in standard form. The quadratic function is a function whose equation includes a variable x 2 (\\ displaystyle x ^ (2)). The equation may include or not include a variable. X (\\ DisplayStyle X). If the equation includes a variable with an indicator of more than 2, it does not describe the quadratic function. If necessary, bring similar members and rear them to record the function in standard form.
A chart of a quadratic function is a parabola. Parabola branches are directed up or down. If the coefficient A (\\ DisplayStyle a) With a variable x 2 (\\ displaystyle x ^ (2)) A (\\ DisplayStyle a)
Calculate -B / 2a. Value - B 2 A (\\ DisplayStyle - (\\ FRAC (B) (2A))) - This is coordinates X (\\ DisplayStyle X) The peaks of parabola. If the quadratic function is recorded in a standard form a x 2 + b x + c (\\ displaystyle ax ^ (2) + bx + c), use the coefficients when X (\\ DisplayStyle X) and x 2 (\\ displaystyle x ^ (2)) in the following way:
The quadratic function is called the function of the form:
y \u003d a * (x ^ 2) + b * x + c,
where a is the coefficient with a senior degree of unknown x,
b - the coefficient at the unknown x,
and with - a free member.
The graph of the quadratic function is a curve called parabola. The general view of the parabola is presented in the figure below.
Fig.1 General view of parabola.
There are several different ways to build a chart of a quadratic function. We will look at the main and most common one.
Algorithm for constructing a graph of a quadratic function y \u003d a * (x ^ 2) + b * x + c
1. Build a coordinate system, note a single segment and sign the coordinate axes.
2. Determine the direction of the branches of the parabola (up or down).
To do this, you need to look at the sign of the coefficient a. If plus - the branches are directed up, if the branches are sent down.
3. Determine the coordinate of the top of the parabola.
To do this, you need to use the Formula of Hvershins \u003d -b / 2 * a.
4. Determine the coordinate at the top of the parabola.
To do this, substitute the alien \u003d a * (x ^ 2) + b * x + c equation instead of x, found in the previous step the value of Hvershina.
5. Apply the resulting point on the chart and spend the axis of symmetry through it, parallel to the OU's coordinate axis.
6. Find the intersection points of the graph with the axis oh.
To do this, solve the square equation A * (x ^ 2) + b * x + c \u003d 0 one of famous methods. If the equation does not have real roots, then the function graph does not cross the axis OH.
7. Find the coordinates of the point of intersection of the graph with the OU axis.
To do this, we substitute the value x \u003d 0 to equation and calculate the value of y. We celebrate this and symmetrical point on the chart.
8. We find the coordinates of arbitrary point A (X, Y)
To do this, select the arbitrary value of the coordinates x, and we substitute it in our equation. We get the value at this point. Apply a point on the chart. And also note the point on the chart, the symmetric point A (x, y).
9. Connect the received points on the smooth line graph and continue the schedule for the extreme points to the end of the coordinate axis. Sign the schedule either on the callout, or if the place is located along the graph.
An example of building a graphic
As an example, we construct a chart of a quadratic function given by the equation y \u003d x ^ 2 + 4 * x-1
1. We draw coordinate axes, we sign them and mark a single segment.
2. The values \u200b\u200bof the coefficients A \u003d 1, B \u003d 4, C \u003d -1. Since a \u003d 1, that more zero branch of parabola is directed up.
3. Determine the coordinate x of the top of the Hvershina parabola \u003d -b / 2 * a \u003d -4 / 2 * 1 \u003d -2.
4. Determine the coordinate at the top of the parabola
Spray \u003d A * (x ^ 2) + b * x + c \u003d 1 * ((- 2) ^ 2) + 4 * (- 2) - 1 \u003d -5.
5. We note the vertex and carry out the symmetry axis.
6. We find the point of intersection of the graph of the quadratic function with the axis oh. We solve the square equation x ^ 2 + 4 * x-1 \u003d 0.
x1 \u003d -2-√3 x2 \u003d -2 + √3. We mark the values \u200b\u200bobtained on the chart.
7. We find the point of intersection of the schedule with the OU axis.
x \u003d 0; y \u003d -1.
8. Select an arbitrary point B. Let it have a coordinate x \u003d 1.
Then y \u003d (1) ^ 2 + 4 * (1) -1 \u003d 4.
9. We connect the points received and subscribe a schedule.
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XIII Regional Scientific Forum of Young Researchers
"Step into the future - 2010"
Research
Quadratic feature: its study and the construction of the schedule.
MOU "Shipakovskaya Main
comprehensive school "
Leader:
Mathematic teacher
MOU "Shipakovskaya Main
comprehensive school "
the Russian Federation
2010
Brief abstract
In this research work, an affordable form contains material about a quadratic function, its properties.
The graphs of 33 of the quadratic functions of different structures were built. On the basis of data, a study algorithm is compiled.
Two ways to build graphs are presented. Defined its algorithm for building graphs.
When writing research work Used published materials, Advanced Grapher, built various graphs. I led my research during the past academic year.
Quadratic function: its study and graphics
Russia, Tyumen region, Yurginsky district, p. Shipakovo,
MOU "Schipakovskaya Main Communication School", 9th grade student.
annotation
Purpose of work:The study of the properties of the quadratic function, features of the location of graphs on the coordinate plane, the study of algorithms for constructing graphs of functions on the coordinate plane.
Tasks:
Explore the properties of the quadratic function. To identify from what the location of the graphs of these functions on the coordinate plane depends. Examine the algorithms for constructing a quadratic function. Learn to quickly and correctly build graphs of quadratic functions on the coordinate plane.
Methods and techniques of work:
Study of graphs of quadratic functions, learning special literature, search for information on the Internet, building charts of quadratic functions using the Advanced Grapher program.
The data obtained:
The location of the graphs of quadratic functions depends on the value of A, B, C, discriminant. You can construct a graph of this function in two ways: by points, in the auxiliary coordinate system through the selection of a full square.
Conclusions:
1.If a \u003d 1, then the graph of the quadratic function is a graph y \u003d x2, transferred parallel to the axis from the top at the point (- ;-).
2.If a\u003e 0, then the branches of parabola are directed upwards. If A.<0, то ветви параболы направлены вниз.
3. All graphics of quadratic functions have an axis of symmetry passing through the top of the parabola, parallel to the axis y, or being.
4. To study the graphs, it is enough to know the value A, the coordinates of the vertices and the intersection points with the x axis.
5.If a \u003d 1, the coordinates of the vertices are integers, it is more convenient to build a graph using the coordinate auxiliary system. If not, then build a schedule by points.
Quadratic function: its study and graphics
Russia, Tyumen region, Yurginsky district, p. Shipakovo,
MOU "Schipakovskaya Main Communication School", 9th grade student.
Research Article
Quadratic function called a function that can be set formula
y \u003d.ax² + bX +.c.where a.≠0.
I decided to build various graphs of quadratic functions using the Advanced Grapher program and explore them. Took arbitrary formulas of quadratic functions, various in structure (formulas of quadratic functions differ from each other with values \u200b\u200bA, B, C). Compared the coordinates of the vertices of parabolla constructed graphs and calculated by the formula (- ; -). And also found the values \u200b\u200bof the discriminant.
1. Quadratic function: y \u003d x2 (Elementary quadratic function: a \u003d 1, b \u003d 0, c \u003d 0). (Attachment 1)
b \u003d 0, C \u003d 0
2. Quadratic function: y \u003d 3x2 (a\u003e 0, b \u003d 0, c \u003d 0) (Appendix 2)
3. Quadratic function: y \u003d -3x2 (and<0, b=0, с=0) (Приложение 3)
4. Quadratic function: y \u003d x2 (0<а <1, b=0, с=0) (Приложение 4)
5. Quadratic function: y \u003d -x2 (0\u003e a\u003e 1, b \u003d 0, c \u003d 0) (Appendix 5)
Graphs of quadratic functions, which a\u003e 0,b \u003d 0.
6. Quadratic function: y \u003d x2 + 4 (a \u003d 1, b \u003d 0, c\u003e 0) (Appendix 6)
7. Quadratic function: y \u003d x2-4 (a \u003d 1, b \u003d 0, with<0) (Приложение 7)
8. Quadratic function: y \u003d 2x2 + 4 (a\u003e 1, b \u003d 0, c\u003e 0) (Appendix 8)
9. Quadratic function: y \u003d 2x2-4 (a\u003e 1, b \u003d 0, with<0) (Приложение 9)
10. Quadratic function: y \u003d x2 + 4 (0<а<1, b=0, с>0) (Appendix 10)
11. Quadratic function: y \u003d x2-4 (0<а<1, b=0, с<0) (Приложение 11)
Graphs of quadratic functions that have a<0, b \u003d 0.
12. Quadratic function: y \u003d - x2 + 5 (a \u003d -1, b \u003d 0, c\u003e 0) (Appendix 12)
13. Quadratic function: y \u003d - x2-5 (a \u003d -1, b \u003d 0, with<0) (Приложение 13)
14. Quadratic function: y \u003d -2x2 + 5 (and<-1, b=0, с>0) (Appendix 14)
15. Quadratic function: y \u003d -2x2-5 (and<-1, b=0, с<0) (Приложение 15)
16. Quadratic function: y \u003d -x2 + 5 (0\u003e A\u003e -1, B \u003d 0, C\u003e 0) (Appendix 16)
17. Quadratic function: y \u003d -x2-5 (0\u003e a\u003e -1, b \u003d 0, with<0) (Приложение 17)
Graphics of quadratic functions thatb.≠0, C \u003d 0
18. Quadratic function: y \u003d x2 + 3x (a \u003d 1, b ≠ 0, c \u003d 0) (Appendix 18)
19. Quadratic function: y \u003d - x2 + 3x (a \u003d 1, b ≠ 0, c \u003d 0) (Appendix 19)
20. Quadratic function: y \u003d 2x2 + 3x (a\u003e 1, b ≠ 0, c \u003d 0) (Appendix 20)
21. Quadratic function: y \u003d -2x2 + 3x (and<-1, b≠0, с=0) (Приложение 21)
22. Quadratic function: y \u003d x2 + 3x (0<а<1, b≠0, с=0) (Приложение 22)
23. Quadratic function: y \u003d -x2 + 3x (0\u003e a\u003e 1, b ≠ 0, c \u003d 0) (Appendix 23)
Graphs of quadratic functions, in which A \u003d 1,b ≠ 0, C ≠ 0
24. Quadratic function: y \u003d x2 + 4x-5 (A\u003e 0, B ≠ 0, C ≠ 0) (Appendix 24)
25. Quadratic function: y \u003d x2 + 4x + 5 (a\u003e 0, b ≠ 0, C ≠ 0) (Appendix 25)
26. Quadratic function: y \u003d x2 + 4x + 4 (A\u003e 0, B ≠ 0, C ≠ 0) (Appendix 26)
Graphs of quadratic functions, whose A \u003d -1,b ≠ 0, C ≠ 0
27. Quadratic function: y \u003d - x2 + 4x + 5 (and<0, b≠0, с≠0) (Приложение 27)
28. Quadratic function: y \u003d - x2-4x-5 (and<0, b≠0, с≠0) (Приложение 28)
29. Quadratic function: y \u003d - x2-4x-4 (and<0, b≠0, с≠0) (Приложение 29)
Graphs of quadratic functions, whose A ≠ 1,b ≠ 0, C ≠ 0
30. Quadratic function: y \u003d 2x2 + 6x + 5 (A\u003e 1, B ≠ 0, C ≠ 0) (Appendix 30)
31. Quadratic function: y \u003d -2x2 + 6x + 5 (and< -1, b≠0, с≠0) (Приложение 31)
Graphs of quadratic functions, from which -1<а<1, b ≠ 0, C ≠ 0
32. Quadratic function: y \u003d x2 + 6x + 15 (0<а <1, b≠0, с≠0) (Приложение 32)
33. Quadratic function: y \u003d -x2 + 6x\u003e a\u003e -1, b ≠ 0, C ≠ 0) (Appendix 33)
The graphs of all quadratic functions is parabola. If A\u003e 0 , then the branches of parabola are directed up. If A.< 0, then the branches of parabola are directed up. Top Parabolia.
y \u003d ah² at point (0; 0); y \u003d ah² + s at point (0; c); y \u003d ah² + q and y \u003d ah² + vx + s at point (- ; -).
Axis of symmetry- This is a straight line relative to which all points of the function schedule are arranged symmetrically. All graphs of quadratic functions have an axis of symmetry passing through the vertex. If the function is specified by the formula y \u003d ah² or y \u003d ah² + s, then the axis of symmetry is the axis y. If the function is defined by the formula y \u003d ah² + bx or y \u003d ah² + BX + C, then the axis of symmetry is straight x \u003d - .
Compression stretching graphs.
Compression: Function Schedule y \u003d AF.(x.) (but \u003e 1) is obtained by stretching the function graphics y \u003d F.(x.) Along the axis y. in but time.
Stretching: Function Schedule y \u003d AF.(x.) (0 < but< 1) получается с помощью сжатия графика функции y \u003d F.(x.) Along the axis y. in time.
The graph of the quadratic functions, at a \u003d 1, is a graph y \u003d x2, transferred parallel to the axis at the vertex (- ;-). If a \u003d -1, then also symmetrically transferred relatively direct y \u003d - (direct passing through the vertex parallel to the x axis).
The graph of quadratic functions at a\u003e 1, regardless of the value B and C, is a graph y \u003d x2, which stretches along the axis of symmetry in but Once from the top, at 0
Dependence The location of the graph of the quadratic function from the discriminant.
The properties of the function and the type of its graph are determined by the value of A and discriminant
D \u003d B.² - 4. aC.
a. > 0, D. > 0 | a. > 0, D. = 0 | a. > 0, D. < 0 |
https://pandia.ru/text/78/547/images/image007_45.gif "Alt \u003d" (! Lang: Parabola1" align="left" width="192 height=187" height="187">!} 
a. < 0, D. > 0 | a. < 0, D. = 0 | a. < 0, D. < 0 |
https://pandia.ru/text/78/547/images/image010_29.jpg "Alt \u003d" (! Lang: Parabola5" width="196" height="177">!} 
Properties of quadratic functions
1. All quadratic functions have a definition area: R, all valid numbers.
2. The range of values \u200b\u200bdepends on the value of A: when a. > 0 [- ; + ∞), when a. < 0 (-∞;- ] .
3. Parity, oddness of quadratic functions: when b. \u003d 0 function is even (i.e. y \u003d ah2 + c \u003d a (s) 2 + s; when b. ≠ 0, then the function is neither even or odd.
4. Zero functions (that is, at what values \u200b\u200bof the argument, the values \u200b\u200bof the function is 0).
If a D. \u003e 0, then the graph of the quadratic function has two zero: x1 \u003d; x2 \u003d.
and the graph of the function crosses the x axis in 2 points.
If a D. \u003d 0, then the graph of the quadratic function has one zero: x. = -;
and the graph of the function concerns the x axis at the point (- ; 0)
If a D. < 0, то график квадратичной функции не имеет нулей, график не пересекает ось х.
5. The intervals of the alternation (gaps from the function of determining the function, where the function takes positive or negative values, i.e.,\u003e 0 or<0).
If a\u003e 0, d\u003e 0, then\u003e 0 at x (-∞; x1) U ( x2; + ∞); W.<0 при хhttps://pandia.ru/text/78/547/images/image014_31.gif" width="13" height="13">(-∞;x.) U ( x.; +∞).
If a\u003e 0, D<0, то у>0 at https://pandia.ru/text/78/547/images/image014_31.gif "width \u003d" 13 "height \u003d" 13 src \u003d "\u003e (x1; x2);<0 при х(-∞;x1) U ( x2; ∞).
If A.<0, D =0, то у<0 при х (-∞;x.) U ( x.; ∞).
If A.<0, D <0, то у<0 при х https://pandia.ru/text/78/547/images/image014_31.gif" width="13" height="13"> [- ; + ∞); decreases at x (-∞; - ].
If A.<0, функция возрастает при х(-∞;- ], decreases at x [- ;+∞).
7. Extremes of the function (maximum points, minimum) at the maximum points (minimum) the value of the function is greater (accordingly less) of all neighboring values.
If a\u003e 0, then the graphs have only minimum of functions, if<0 – только максимум функций. Это точки вершины параболы.
If a a. \u003e 0, then x.mIN \u003d - ; y.mIN \u003d - ; if a a. < 0 x.max \u003d -; y.max \u003d -.
Algorithm for study of the properties of the quadratic function
Domain. Area of \u200b\u200bvalues. Parity odd function. Zero function. Gaps of alpopurism. Monotonicity gaps. Extreme function.
After analyzing the construction of the graphs of my quadratic functions, I am algorithm for constructing graphs of quadratic functions by points (1 method).
We find the abscissa of the pearabol vertex according to the formula x0 \u003d - . We find the value of U0 according to the formula U0 \u003d - . On the coordinate plane we build a pearabol vertex with coordinates (x0; u0). We define the direction of the branches of the parabola (according to the coefficient a). We will carry out the axis of symmetry of the parabola through its vertex, parallel to the axis y. Select the values \u200b\u200bof the left or right of the axis of the parabola symmetry and fills the values \u200b\u200btable. We build points along the coordinates obtained on the coordinate plane. We build a chart of a quadratic function without restrictions on extreme points and subscribe a schedule.
I build on this algorithm graph y \u003d x2 - 4x + 3
2. d \u003d b2-4As \u003d (- \u003d 4 y \u003d - = .
4. A\u003e 0, Parabola branches are directed up.
5. Symmetry axis straight x \u003d 2.
6. Table of values
7.Tractive points with the coordinates obtained on the coordinate plane.
8th grade "HREF \u003d" / "REL \u003d" BOOKMARK "\u003e 8 Class We learned to allocate full square in square equations. Elena Nikolaevna even then said that it depends on the location of the schedule on the coordinate plane. I decided to check : It is possible through the allocation of a complete square to make an algorithm for constructing graphs of quadratic functions on the coordinate plane.
The equations of my quadratic functions were examined from 18-33 and compared the obtained formulas with the vertices of the built graphs:
18. y \u003d x2 + 3x \u003d (x2 + 2 · 1.5 · x +2.25) - 2,25 \u003d (x + 1.5) 2-2.25 a \u003d 1 vertex (-1,5;-2,25)
19. y \u003d - x2 + 3x \u003d -1 (x2-2 · 1.5 · x +2.25) + 2,25 \u003d -1 (x - 1.5) 2 +2,25 A \u003d -1 Top (1,5; 2,25)
20. y \u003d 2x2 + 3x \u003d 2 (x2 + 2 · 0.75 · x + 0.5625) -1,125 \u003d 2 (x + 0.75) 2 -1.125 A \u003d 2.
vertex (-0,75;-1,125)
21. y \u003d -2x2 + 3x \u003d -2 (x2-2 · 0.75 · x +0.5625) +1,125 \u003d -2 (x-0,75) 2 +1,125 A \u003d 2.
vertex (0,75;1,125)
22. y \u003d x2 + 3x \u003d (x2 + 2 · 3 · x + 9) - 4,5 \u003d (x +3) 2 -4.5 a \u003d https: //pandia.ru/text/78/547/images/image004_61.gif "width \u003d" 16 height \u003d 41 "height \u003d" 41 "\u003e x2 + 3x \u003d - (x2 -2 · 3 · x + 9) + 4.5 \u003d - (x -3) 2 +4.5 A \u003d -HTTPS: //pandia.ru/text/78/547/images/image004_61.gif "width \u003d" 16 height \u003d 41 "height \u003d" 41 "\u003e x2 + 4x + 15 \u003d (x2 + 2 · 6 · x + 36) -18 + 15 \u003d (x +6) 2 -3 A \u003d https: //pandia.ru/text/78/547/images/image004_61.gif "width \u003d" 16 height \u003d 41 "height \u003d" 41 "\u003e x2 + 6x-14 \u003d - (x2 -2 · 6 · x + 36) +18 -14 \u003d - (x -6) 2 +4 a \u003d https: //pandia.ru/text/78/547/images/image001_112.gif "width \u003d" 24 "height \u003d" 41 "\u003e; n \u003d - . That is the coordinates of the vertex parabola (m; n)
The algorithm for constructing a chart of a quadratic function using an auxiliary coordinate system through the allocation of a full square (2 method).
1. Reformation of formula y \u003d ah² + vx + c \u003dy \u003d a (x -m) 2 +n.where m \u003d - ; n \u003d -
or y \u003d a (x +) 2 -
2. Stretching graphics y \u003d X.2 along the axis w. in but
Once at a\u003e 1, at 0< a. < 1 - это сжатие в a. time.
If a a.< 0, произвести ещё и зеркальное отражение графика относительно оси h. (Parabola branches will be directed down).
Conversion Result: Function Schedule y \u003d ax.2.
https://pandia.ru/text/78/547/images/image020_21.jpg "width \u003d" 147 "height \u003d" 193 src \u003d "\u003e
y \u003d A.(x - M.) 2 along the axis y. on n (up n. \u003e 0 and down when n. < 0). Результат преобразования: график функции y \u003d A (X-M)2 + N.
https://pandia.ru/text/78/547/images/image026_15.jpg "width \u003d" 336 "height \u003d" 161 src \u003d "\u003e
4. Parallel transfer of function graphics
y. = - (x. + 2) 2 along the axis y. on -1.
Grade 6 "HREF \u003d" / TEXT / CATEGORY / 6_KLASS / "REL \u003d" BOOKMARK "\u003e 6th grade ,. - ed.4 - M. Publishing House" Russian Word ", 1997" Algebra ". Tutorial grade 9. ,. M. Enlightenment, 2004 "Mathematics" Weekly educational and methodical newspaper. Publishing house "FIRST SEPTHER". No. 48, 2003 "Mathematics" Weekly educational and methodical newspaper. Publishing House "First September". №7, 1998 Tests and exam tasks in mathematics. Tutorial. . - Publishing House "Peter", 2005 "Absolute Value". . - M.: Enlightenment, 1968. "Functions and Graphic Construction". .- M.: Enlightenment, 1968. "The tasks of increased difficulty in the course of algebra for 7-9 classes." . M.: Education, 1991.
Quadratic function: its study and graphics
Russia, Tyumen region, Yurginsky district, p. Shipakovo,
MOU "Schipakovskaya Main Communication School", 9th grade student.
Research Plan
Justification of the problem. In control and measuring materials on algebra in grade 9 for the passage of state final certification in new form It turned out that many tasks are found to build graphs of quadratic functions, their research. When constructing graphs of a quadratic function, difficulties arise due to the fact that when drawing up a table of values \u200b\u200bwith small, modulo, the values \u200b\u200bof the argument, the function values \u200b\u200bare sometimes very large, modulo, and are not included on the notebook page. Therefore, I decided to explore: the properties of the quadratic function and the location of the graphs of the quadratic functions on the coordinate plane depends; Examine algorithms for building charts of these functions and choose the most light algorithm for constructing a quadratic function.
Hypothesis:
If I study the properties of a quadratic function, the graph of graphic algorithms, I detect, from which the location of the graphs on the coordinate plane depends, then I can quickly and correctly build graphs of this feature by choosing the most easy way construction; Explore this feature.
Description of the method:
1. Analyzing its quadratic functions, I concluded that it is enough to know for studying the properties of functions:
Value A: To determine the directions of parabola branches, compression and stretching of graphs, gaps of the alignment;
The coordinates of the vertices of Parabola: to determine the range of values, intervals of monotony, extremes of the function;
Value B: To determine parity, or no parity, neitherness;
The value of the discriminant: to determine the number of zeros of functions;
If D< 0, то нулей функции нет;
If d \u003d 0, then the zero function is one - this is the top of the parabola;
If D\u003e 0, then zeros of function 2.
Zeros of functions: to determine the intervals of the alignment.
2. Working on its topic, I brought my own way of building charts of a quadratic function (using the coordinate auxiliary system) according to the following algorithm:
- Determine the peaks of parabola. Build an auxiliary coordinate system with a center at a vertex point. Build a graph y \u003d x2 at the point of the vertex if a\u003e 0, then the branches refer up.
If and \u003c0, then the branches will be down.
- If Iai\u003e 1, then stretch the schedule with respect to the axis of symmetry in and once
If 0 \u003ciai \u003c1, then squeeze the graph with respect to the axis of symmetry in and once
3. Construction of graphs of quadratic functions is convenient to carry out different ways. If a \u003d 1, the coordinates of the vertices are integers, then with the help of the coordinate system. If a ≠ 1, the coordinates of the pearabol vertices are not integers, then the way: by points.
4. In the lessons of algebra in grade 9 after performing this research work, I help classmates to assimilate these methods for building charts of quadratic functions with my ways, to conduct their research.
Result:
During the research work, I am the algorithm for studying the properties of the quadratic function and tested it in practice. I learned that the quadratic functions can be set in two ways: AH2 + BX + C and A (X-M) + N. He learned from 2 algorithms to build graphs of these functions. I revealed from which the location of graphs on the coordinate plane depends. Created toolkit "Submarine stones of a quadratic function", which distributed to his school students, presented to other schools. In the future, I plan to explore quadratic functions that have a module in the formula.
The tasks for properties and graphs of the quadratic function cause, as practice shows, serious difficulties. It is rather strange, because the quadratic function is held in the 8th grade, and then the entire first quarter of the 9th grade "survive" the properties of the parabola and build its graphs for various parameters.
This is due to the fact that forcing students to build parabolas, almost do not pay time for reading charts, that is, not practicing the understanding of the information obtained from the picture. Apparently, it is assumed that by building a dozen two charts, a smart schoolboy will detect himself and formulates the connection of coefficients in the formula and appearance graphics. In practice it does not work. For such a generalization, a serious experience of mathematical mini studies, which most nine-graduates, of course, do not have it. Meanwhile, in GIA suggest precisely on the schedule to determine the signs of coefficients.
Let's not require schoolchildren impossible and simply offer one of the algorithms to solve such problems.
So, the function of the form y \u003d AX 2 + BX + C It is called a quadratic, the schedule is parabola. As follows from the name, the main term is aX 2.. I.e but should not be zero, the remaining coefficients ( b. and from) can be zero.
Let's see how the signs of its coefficients affect the appearance of the parabola.
The simplest dependence for the coefficient but. Most schoolchildren confidently replies: "If but \u003e 0, then the parabola branches are directed upwards, and if but < 0, - то вниз". Совершенно верно. Ниже приведен график квадратичной функции, у которой but > 0.
y \u003d 0.5X 2 - 3X + 1
In this case but = 0,5
And now for but < 0:
y \u003d - 0.5x2 - 3x + 1
In this case but = - 0,5
Influence of the coefficient from Also easy to trace enough. Imagine that we want to find the value of the function at the point h. \u003d 0. Substitute zero in the formula:
y. = a. 0 2 + b. 0 + c. = c.. Turns out that y \u003d s. I.e from - This is the ordinate of the point of intersection of the parabola with the axis. As a rule, this point is easy to find on the chart. And determine above zero it lies or below. I.e from \u003e 0 or from < 0.
from > 0:
y \u003d x 2 + 4x + 3
from < 0
y \u003d x 2 + 4x - 3
Accordingly, if from \u003d 0, then Parabola will definitely pass through the origin of the coordinate:
y \u003d x 2 + 4x
More difficult with the parameter b.. The point on which we will find it depends not only from b. But from but. This is the top of the parabola. Its abscissa (axis coordinate h.) is on the formula x B \u003d - b / (2a). In this way, b \u003d - 2ach in. That is, we act as follows: on the chart we find the top of the parabola, we define the sign of its abscissa, that is, we look to the right of zero ( x B. \u003e 0) or left ( x B. < 0) она лежит.
However, this is not all. We also need to pay attention to the coefficient sign but. That is, to see where the branches of parabola are directed. And only after that by the formula b \u003d - 2ach in Determine the sign b..
Consider an example:
Branches are directed up, it means but \u003e 0, Parabola crosses the axis w. below zero, then from < 0, вершина параболы лежит правее нуля. Следовательно, x B. \u003e 0. So b \u003d - 2ach in = -++ = -. b. < 0. Окончательно имеем: but > 0, b. < 0, from < 0.
