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It is known that the quadratic function y = AX2 + BX + C (a ≠ 0) satisfies the following quantitative relationship between the independent variable x and the function value y X negative two thirds negative one negative half 0 half 1 half three Four fifths of Y minus two minus nine fourths of Y minus two minus five fourths of Y minus seven fourths of Y
Select three groups of values (- 1, - 2); (0, - 2); (1,0) and substitute them respectively (because these numbers are convenient for calculation)
a-b+c=-2
c=-2
a+b+c=0
The results of solving equations
a=1
b=1
c=-2
The analytic formula is y = x ^ 2 + X-2
The partial corresponding values of the independent variable X of the quadratic function y ax 2 + BX + C and the function value y are as follows: X... - 2 - 102 t 5... Y... - 7
x...-2 -1 0 2 t 5 ...
y...-7 -2 1 1 -7 -14...
(1) (1) t = - (2) maximum value of quadratic function (3) if points a (x1, Y1) and B (X2, Y2) are two points on the image
And - 1 ＜ x 1 ＜ 0,4 ＜ x 2 ＜ 5 is the comparison of Y 1 and y 2
(2) Find the root of ax & # 178; + BX + C = 0——
(3) If the value range of independent variable x is - 3 ≤ x ≤ 3, then the value range of function value y is -
4a-2b+c=-7
a-b+c=-2
c=1
So the solution is a = - 1, B = 2, C = 1
By x = 0,2, y = 1
We obtain that the axis of symmetry is x = 1, and if x = - 2, y = - 7
So t = 4
Maximum value of quadratic function = 2
y2＜y1
2. Root formula 1 + 2 ^ 0.5 1-2 ^ 0.5
The root number cannot be typed
3.-14→2
RELATED INFORMATIONS
- 1. It is known that in the quadratic function y = AX2 + BX + C, the corresponding values of function y and independent variable X are shown in the following table X - 3 - 20135 y 708 - 9 - 57 find the function value y when the symmetry axis X = 2 of y = ax + BX + C
- 2. If the point P (a, √ 3) is on the positive scale function y = - 3x, find the value of A
- 3. Y = (m-2) x to the power of M + 1 + 3x is a positive proportion function, and finding the relationship of positive proportion function is a formula
- 4. It is known that the quadratic function f (x) = ax Λ 2 + BX + 1 (a, B are constants, a ＞ 0) satisfies f (x-3) = f (1-x) and has the maximum value 4 in the interval ＜ - 3,2 ＞ find the value of a, B and find the range of the function f (x) in the interval ＜ - 3,2 ＞
- 5. The quadratic function f (x) = ax ^ 2 + BX + C satisfies the following conditions 1) F (- 1) = f (- 3); 2) f (x) has a minimum value of 3; 3) f (x) image passes through point (0,4), find the analytic expression of function f (x)
- 6. It is known that the left focus of the ellipse x ^ 2 + y ^ 2 / b ^ 2 = 1 (b belongs to (0,1)) is f, the left and right vertices are AC, the upper vertex is B, and the circle P is made through three points of FBC, where the coordinates of the center P are (m, n). Let f, B and C be (- C, 0), (0, b), (1,0) respectively. If the eccentricity of the ellipse e = 2 / 3, the equation of the circle P is obtained
- 7. It is known that the right focus a of ellipse C: x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0) is the focus of parabola y ^ 2 = 8x, the vertex on the circle is B, and the eccentricity is √ 3 / 2,1, Find the path of ellipse C; 2, the line L passing through point (0, √ 2) with slope k intersects ellipse C at two points P and Q. if the abscissa of the midpoint of line PQ is 4 √ 2 / 5, find the equation of line L
- 8. The center of a circle is on the right focus of the ellipse 16x ^ 2 + 25y ^ 2 = 400, and passes through the vertex of the ellipse on the y-axis to find the equation of the circle
- 9. The two ends of the minor axis of the ellipse form 120 degrees with the line connecting the focus, and the eccentricity of the ellipse is calculated
- 10. From an end point of the ellipse minor axis to see the two focal points, the angle of view is 120 to calculate the eccentricity
- 11. The image of the linear function Y1 = KX + B and the inverse scale function y2 = m / X intersect at two points a (- 2,1), B (1, n) 1) Find the expression of first-order function and inverse proportion function 2) Write Y1 > Y2
- 12. The function f (x) = X3 + AX2 + BX + C, where a, B and C are real numbers. When a2-3b < 0, f (x) is () A. Increasing function B. decreasing function C. constant D. is neither increasing function nor decreasing function
- 13. It is known that the minimum value of function f (x) = (2 ^ x-a) ^ 2 + (2 ^ - x-a) ^ 2 is 8, and there is a value range of real number a
- 14. It is known that a and B are real numbers, f (x) = x ^ 3 + ax, G (x) = x ^ 2 + BX, f '(x) and G' (x) are derivatives of F (x), G (x), if f '(x) g' (x) ≥ 0 in function area If f (x) and G (x) are monotonically consistent on interval I, then f (x) and G (x) are monotonically consistent on interval I 1. Let a > 0, if the monotonicity of functions f (x) and G (x) is consistent in the interval [- 1, + ∞), find the value range of real number B
- 15. Let the asymptote of hyperbola x2a2 − y2b2 = 1 (a > 0, b > 0) be tangent to the parabola y = x2 + 1, then the eccentricity of the hyperbola is equal to () A. 3B. 2C. 5D. 6
- 16. It is known that a focus of hyperbola C: x ^ 2 / A ^ 2-y ^ 2 / b ^ 2 = 1 (a > 0, b > 0) is F2 (2,0) and B = root 3a. (1) the equation for solving hyperbola C (2) let a normal vector of line L passing through focus F2 be (m, 1). When line L and the right branch of hyperbola C intersect at two different points a and B, then, Find the value range of the real number m: and prove that the key point m of AB is on the curve 3 (x-1) ^ 2-y ^ 2 = 3. (3) let the right branch of the straight line L and the hyperbola C in (2) intersect at two points a and B. ask if there is a real number m, so that the angle AOB is an acute angle? If there is, find the range of M. if not, explain the reason
- 17. Polar coordinate equation 2 ρ cos ^ 2 θ / 2 = 5 is a circle B ellipse C hyperbola D parabola
- 18. Given that the polar equation of the parabola is ρ = 41 − cos θ, then the Quasilinear polar equation of the parabola is ρ = 41 − cos θ______ ．
- 19. If the polar coordinate equation of the curve is ρ & # 178; = 400 / (16cos & # 178; θ + 25sin & # 178; θ), the general equation of the curve is?
- 20. It is known that the parametric equation of C is x = 3cost y = 3sint (t is the parameter), and the tangent of C at point (0,3) is o If the origin of angular coordinate is the pole and the positive half axis of X axis is the polar axis, the polar coordinate equation of O is?