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On all concepts of quadratic function
Quadratic function knowledge points and related typical topics
What is the standard expression of quadratic function?
f(x)=ax^2+bx+c
(a ≠ 0, B and C are arbitrary constants)
Quadratic function problem on line, etc
Given that the quadratic function f (x) = x ^ 2 + BX + C passes through P (1,0), and for any real number x, f (1 + x) = f (1-x), the maximum value of F (x) is obtained
It's a process. Thank you!
Substituting (1,0) into f (x) = x ^ 2 + BX + C, we get:
1+b+c=0
c=-b-1
(1+x)^2+b(1+x)+c=(1-x)^2+b(1-x)+c
(2b+4)x=0
2b+4=0
b=-2
c=-(-2)-1=1
f(x)=x^2-2x+1=(x-1)^2
Therefore, f (x) has a minimum value of 0
It is known that the intersection points of the image and the coordinate axis of the quadratic function y = ax & sup2; + BX + C are respectively a (- 1,0) B (3,0) C (0, - 3), and the maximum value of this function is obtained
In general, the relationship between independent variable x and dependent variable y is as follows:
Y = ax ^ 2 + BX + C (a, B, C are constants, a ≠ 0, and a determines the opening direction of the function. When A0, the opening direction is upward, and when A0, the opening direction is downward. IAI can also determine the opening size. The larger IAI is, the smaller the opening is, and the smaller IAI is, the larger the opening is.)
Then y is called the quadratic function of X
The right side of quadratic function expression is usually quadratic trinomial
2. Three expressions of quadratic function
General formula: y = ax ^ 2; + BX + C (a, B, C are constants, a ≠ 0)
Vertex formula: y = a (X-H) ^ 2; + K [vertex P (h, K)]
Intersection formula: y = a (x-x1) (x-x2) [only parabola with intersection a (x1,0) and B (x2,0) with X axis]
Note: in the mutual transformation of the three forms, there are the following relations:
h=-b/2a k=(4ac-b^2;)/4a x1,x2=(-b±√b^2;-4ac)/2a
3. The image of quadratic function
The image of quadratic function y = X2 is made in plane rectangular coordinate system,
It can be seen that the image of quadratic function is a parabola
4. Properties of parabola
1. Parabola is an axisymmetric figure. The axis of symmetry is a straight line
x = -b/2a.
The only point of intersection between the axis of symmetry and the parabola is the vertex P of the parabola
In particular, when B = 0, the symmetry axis of the parabola is Y-axis (that is, the line x = 0)
2. The parabola has a vertex P whose coordinates are
P [ -b/2a ,(4ac-b^2;)/4a ].
When - B / 2A = 0, P is on the y-axis; when Δ = B ^ 2-4ac = 0, P is on the x-axis
3. The quadratic coefficient a determines the opening direction and size of the parabola
When a > 0, the parabola opens up; when a < 0, the parabola opens down
|The larger a | is, the smaller the opening of the parabola is
4. The position of the axis of symmetry is determined by the coefficient b of the first term and the coefficient a of the second term
When a and B have the same sign (AB > 0), the symmetry axis is on the left of Y axis;
When a is different from B (AB < 0), the axis of symmetry is on the right of Y axis
5. The constant term C determines the intersection of the parabola and the y-axis
The intersection of parabola and y-axis at (0, c)
6. The number of intersections of parabola and x-axis
When Δ = B ^ 2-4ac ＞ 0, there are two intersections between the parabola and X axis
When Δ = B ^ 2-4ac = 0, there is one intersection point between the parabola and X axis
When Δ = B ^ 2-4ac ＜ 0, there is no intersection between parabola and X axis
5. Quadratic function and quadratic equation of one variable
In particular, the quadratic function y = ax ^ 2; + BX + C,
When y = 0, the quadratic function is a quadratic equation of one variable with respect to X,
That is ax ^ 2; + BX + C = 0
At this point, the function image and X-axis intersection, that is, whether the equation has real roots
The abscissa of the intersection of function and X axis is the root of the equation
2008-07-05 21:36
L Δ o Δ ve ＆ comments on 2o08
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Tag: function property dependent variable other answers
Parabola, axis of symmetry
2008-07-06 19:37
1. The function is called quadratic function, which uses multimedia to demonstrate the influence of the change of parameters,, and on the function image, focusing on the demonstration of the influence on the function image
2. Through the following aspects to study the function
(1) , formula
(2) Find the intersection of function image and coordinate axis
(3) Symmetry properties of functions
(4) Monotonicity of functions
3. Example: study the image and property of function
(1) Formula
Therefore, the image of function can be regarded as a result of a series of transformations. Specifically, the abscissa of each point on the function is changed to twice the original, and then the image is moved 4 units to the left and 2 units to the down
(2) The intersection of function and X axis is (- 6,0) and (- 2,0), and the intersection of function and Y axis is (0,6)
(3) The axis of symmetry of a function is x = - 4. In fact, if a function satisfies: (), then the function is symmetric
(4) Let,,
= =
=
Because,
therefore
If the function y = (A-1) x ^ (B + 1) + x ^ 2 + 1 is a quadratic function, try to discuss the value range of a and B
The third answer on the first floor should be that a is not equal to 1,
Because the function y = (A-1) x ^ (B + 1) + x ^ 2 + 1 is quadratic
So the exponent of X is B + 1 = 0 or B + 1 = 1 or B + 1 = 2
When B + 1 = 0, that is, B = - 1, A-1 is any real number, that is, a is any real number
When B + 1 = 1, that is, B = 0, A-1 is any real number, that is, a is any real number
When B + 1 = 2, that is, B = 1, A-1 ≠ - 1, so a ≠ 0
Know the value range of X in quadratic function and find the value range of quadratic function
When x is greater than or equal to 0 and less than or equal to 3, the value range of function value y of quadratic function y = 3x ^ 2-12x + 5 is
If it's converted to vertex form, it's 3 (X-2) ^ 2-7
Good! You've turned it into a vertex form. You can see that the axis of symmetry of this function is x = 2, opening up. The left side of the axis of symmetry is a decreasing function, and the right side is an increasing function
Substituting x = 0 into y = 5, x = 2, y = - 7, x = 3, y = - 4 - 7 ≤ y ≤ 5, respectively
Given that the quadratic function f (x) = x ^ 2 - (A-1) x + 5 is an increasing function in the interval (1 / 2,1), try to find the value range of F (x)
Wrong. Try to find the range of F (2)
Axis of symmetry x = (A-1) / 2
Opening up
So on the right side of the axis of symmetry is an increasing function
So (A-1) / 2 = 15
Given the quadratic function y = - 3 (x-1) (x + 2), let y = - 3 (x-1) (x + 2)
You can get it by drawing
X greater than 1 or X less than - 2
This kind of problem is error prone and must not be brought into the calculation
If the function value of quadratic function y > 0, the value range of independent variable x is a < x < B, then when y < 0, the value range of X is ()
X < A or x > b
Because when y > 0, X is between a and B, then we can see that the opening of the function is downward and a < B, intersects with the X axis, and draw a rough image with a and B to get the answer
Quadratic function y = ax ^ 2 (a > 0), try to compare the size of y value corresponding to x = 3, x = - Π, x = 4, and find the value range of y when - 1 ≤ x ≤ 3
Quadratic function y = ax ^ 2 (a > 0)
Its symmetry axis is x = 0, that is, Y axis
A > 0
So when x > 0, this function is an increasing function
Therefore, when x = 4, the value of Y is the largest, and when x = 3, the value of Y is the smallest
When - 1 ≤ x ≤ 3, when x = 0, it has a minimum value, and its value is y = 0
Because 3-0 > - 1-0, that is, when x = 3, it has the maximum value, and its value is y = a * 3 ^ 2 = 9A
So the value range of Y is 0 ≤ y ≤ 9A
RELATED INFORMATIONS
- 1. ABC problem of quadratic function The teacher said ABC is a constant Y = ax ^ 2 + BX + C (a, B, C are constants, a ≠ 0, and a determines the opening direction of the function. When a > 0, the opening direction is upward, a ≠ 0
- 2. When x = 3, the minimum value of y = - 1, and the image is over (0,7), the analytic expression of quadratic function about X is obtained
- 3. Some junior high school English exercises 1.Our principal said that there_____ a school meeting the next Monday? A.would have B,was going to be C.will be D,was going to have 2.-----Why dosen't Jack have his computer repaired? ----I know _______ .Perhaps he wants to buy a new one. A,why does he B,why he doesn't C,why he does D,wht doesn't he 3.To see is _____ . A,believing B,believe C,believes D,to believe 4.If she can't come,we need to find_________ . A,someone else B.someone others C,anyone other D,anyone else 5.A person _____ good manners never laughs at_______ A,in;other B,with;others C,with;the other D,in;others 6.Every spring,many trees_______ along the river. A,were planted B,is planted C,will be planted D,are planted 7.I met them ____ the morning of Teachers' Day. A,in B,on C,at D,to Mary had the ambition to be a scientist in her youth Mary ____ ____ _____ ______ being a scientist in her youth. 9.Mike said,"These books are mine." Mike said_____ books ______ with me.
- 4. Analysis of typical examples of quadratic function Quadratic function y = AX2 + BX + C (a is not equal to 0), where a, B, C satisfy a + B + C = 0 and 9a-3b + C = 0, the symmetry axis of quadratic function image is a straight line
- 5. The problem of the maximum value of quadratic function in the first grade of Higher Education 1. Find the maximum value of the function y = x Λ 2 + 2aX + 1, - 2 ＜ = x ＜ = 1. 2. Find the maximum value of the function y = x Λ 2 + 2x-1, m ＜ = x ＜ = m + 1. 3. Find the maximum value of the function y = - x Λ 2 + 2aX + 1-A, 0 ＜ = x ＜ = 1 is 2, and find a (x Λ 2 represents the square of x) (the answer should be detailed)
- 6. Isosceles trapezoid ABCD, AD / / BC, ab = DC, P is any point of BC, PM / / CD, intersects BD with m, PN / / AB intersects AC with N, what is the relationship between PM + PN and ab?
- 7. If two quadratic functions, images and shapes are the same, are their a equal?
- 8. In order to find the image property of quadratic function y = a [X-H] ², What are the symmetry axis, vertex coordinates and maximum of y = a [X-H] &# 178;? I don't want the property of y = a [X-H] &# 178; + K. thank you
- 9. What properties can be summarized from the images of quadratic functions y = x ^ 2 and y = - x ^ 2?
- 10. In quadratic function, why is the larger | a | the smaller the opening of parabola?
- 11. Find the analytic expression of quadratic function satisfying the following conditions: (1) the image passes through points a (0,3), B (1,3), C (- 1,1); (2) the image passes through points a (- 1,0), B (3,0), C (0,6); the image vertex coordinates are (1, - 6), and pass through points (2, - 8) Prawn, tears run ah, math children really tears ah, you can't afford to hurt ah, QAQ!
- 12. Write down the coefficient of quadratic term, coefficient of first term and constant term of quadratic function Coefficient of quadratic term and constant term of coefficient of first term in analytic expression of function y=x^2+2x-1 y=x^2 y=-3x^2+2 y=1/3(x-5)^2-4
- 13. There is an image of a quadratic function. Three students have described some of its characteristics: A: the axis of symmetry is a straight line x = 4; B: the abscissa of the two intersections with the X axis is an integer, and the ordinate of the intersection with the Y axis is also an integer; C: the area product of the triangle with the three intersections as the vertex is 12______ ．
- 14. An image problem of quadratic function that bothers me for many years No matter m is any real number, the image of quadratic function y = x * x - (2-m) x + m always passes through (,) The answer is (- 1,3). I don't know how
- 15. Find the analytic expression of this quadratic function When the input value is x, the input value is y, and Y is a quadratic function of X. when the input value is - 2,0,1, the corresponding output values are 5, - 3, - 4, respectively. Find the analytic expression of the quadratic function
- 16. The image of quadratic function can be solved by (1.6) (- 1. - 18) (2.9)
- 17. On the derivation of double radical of quadratic function Using the original method, find its derivation steps
- 18. As shown in the figure, point P is the inverse scale function y = K / X (k)
- 19. How to draw an angle equal to a known angle with a ruler and a compass
- 20. Drawing conic curve with Zenyang Geometer's Sketchpad