Let the image of two quadratic functions be symmetric with respect to the straight line x = 1, and the analytic expression of one function is y = x ^ 2 + 2x + 1?
Let a point coordinate on another function be (x, y), then its symmetric point coordinate is: (2-x, y), and because this symmetric point is on the first function, it has:
y=(2-x)^2+2(2-x)+1=4-4x+x^2+4-2x+1=x^2-6x+9
The other is y = x ^ 2-6x + 9
RELATED INFORMATIONS
- 1. It is known that the image of F (x) = 2 ^ x, F1 (x), F2 (x), F3 (x), F4 (x) and the image of F (x) are symmetric about the X axis, Y axis, origin and line y = x respectively Try to write four function analytic expressions respectively
- 2. If the domain of F (x) is symmetric about the origin, then F1 (x) = f (x) + F (- x) is an even function and F2 (x) = f (x) - f (- x) is an odd function
- 3. F (x) = 1 + X / 1-x, F1 (x) = f (x), FK + 1 (x) = f [FK (x)], k = 1,2..., find F3 (x), F4 (x), F5 (x) I figured out that F1 (x) = 1 + X / 1-x, F2 (x) = - 1 / x, but F3 (x), F4 (x), F5 (x) are not big
- 4. If the even function y = f (x) decreases monotonically in the interval [0,4], then a f (- 1) > F (one third) > F (- 1) > b f (one third) > F (- 1) > F (- 1) > C f (- 1) > F (one third) > d f (- 1) > F (- 1) > F (one third) > F (one third)
- 5. Among the following functions, the function which is both even and monotonically decreasing on the interval (0,1) is () A. y=1xB. y=lgxC. y=cosxD. y=x2
- 6. Among the following functions, the function that is both even and monotonically increasing on the interval (0, + infinity) is A,y=x∧-1.B,y=log2x.C,y=|x| D,y=-x∧2
- 7. The even function y = f (x) decreases monotonically in the interval [0,4]. Compare the sizes of F (- 1), f (3 / π), f (- π)
- 8. Y = f (x) is defined in R, and its graph is symmetric with respect to the straight line x = A and x = B (a is not equal to b). It is proved that f (x) is a periodic function
- 9. If f (x) is defined as a function on R, and f (10 + x) = f (10-x), f (20-x) = - f (20 + x), it is proved that f (x) is an odd function and periodic function
- 10. On the graph of the function y = 3 ^ X and y = Log1 / 3 1 / X_ symmetric On the graph of the function y = 3 ^ X and y = Log1 / 3 1 / X____ symmetric
- 11. If the image of functions g (x) and f (x) = - 2x + 1 is symmetric with respect to y = x, then G (x)=___ Ask for detailed explanation
- 12. Given the function f (x) = sin (Wx + π / 4), where w > 0, if the distance between two adjacent symmetrical axes of the function f (x) image is equal to π / 3, the analytic expression of the function is obtained And find the minimum positive real number m so that the function corresponding to m unit length of the function image is even
- 13. It is known that ω > 0,0
- 14. Known w > 0,0
- 15. Given that the function y = 2Sin (Wx + Fei) is an even function (w > 0,0 < Fei < π), the distance between two adjacent symmetry axes of an image is π 2, and f (π 9) is obtained After the image of the function is shifted to the right by π - 6 units, the decreasing interval of the function is obtained
- 16. The known function f (x) = 2Sin (Wx + φ) (W > 0, - π / 2)
- 17. It is known that y = 2Sin (Wx + a-pai / 6) 0
- 18. f(x)=x/(x^2+1) Does the equation f (x) - (x-1) / x = 0 have a root? If there is a root x0, ask for an interval (a, b) of length 1 / 4, so that XO ∈ (a, b). If not, explain the reason?
- 19. Given the function f (x) = 2Sin - (2x - π / 6), 1. Write the equation of symmetry axis, symmetry center and monotone interval of function f (x). 2. Find the maximum and minimum value of function f (x) in the interval [0, π / 2]
- 20. (related functions) Given the set a = {Y / y = x Square-1, X contained in R}, B = {X / y = root 2X-4}, then the intersection of a and B = -, the union of a and B=—— The result is {X / X is greater than or equal to 2}, and the next lattice is {X / X is greater than or equal to - 1}