F (x) = 2cos ^ 2 Wx + 2Sin Wx cos Wx + 1 (x ∈ R W ＞ 0) 1 find the value of W and find the symmetric center and axis equation of the function

F (x) = 2cos ^ 2 Wx + 2Sin Wx cos Wx + 1 (x ∈ R W ＞ 0) 1 find the value of W and find the symmetric center and axis equation of the function

f（x）=2cos^2 wx+2sin wx cos wx+1
=1+cos2wx+sin2wx+1
=2+cos2wx+sin2wx
=2+√2[(√2/2)sin2wx+(√2/2)cos2wx]
=2+√2[sin2wxcosπ/4+cos2wxsinπ/4]
=2+√2sin(2wx+π/4)
The value of W cannot be determined because there is a difference condition;
Symmetry center: (K π / (2W) - π / (8W), 2);
Symmetry axis equation: x = k π / (2W) + π / (8W)

The known function f (x) = 1 + 2Sin (2 ω x + π) 6) If the line x = π 3 is an axis of symmetry of the graph of function f (x) (1) Find ω and the minimum positive period (2) Find the monotone decreasing interval of function f (x), X ∈ [- π, π]

(1) It can be seen from the title: 2 ω· π
3+π
6＝kπ+π
2 (K ∈ z), so we have ω = 1
2+3
2k．
And ∵ 0 ᙽ ω ᙽ 1,

Find the value range, monotone interval, symmetry axis and symmetric point of the function y = 2Sin (2x - π / 6),

1. When 2x - π / 6 = π / 2 + 2K π, that is, when x = π / 3 + K π, (K ∈ z), ymax = 2 × 1 = 2,
When 2x - π / 6 = 3 π / 2 + 2K π, i.e. x = 5 π / 6 + K π, (K ∈ z), Ymin = 2 × (- 1) = - 2, νrange [- 2,2]
2. From - π / 2 + 2K π ≤ 2x - π / 6 ≤ π / 2 + 2K π, we get - π / 6 + K π ≤ x ≤ π / 3 + K π, K ∈ Z,
The single increasing interval is [- π / 6 + K π, π / 3 + K π], K ∈ Z;
From π / 2 + 2K π ≤ 2x - π / 6 ≤ 3 π / 2 + 2K π, π / 3 + K π ≤ x ≤ 5 π / 6 + K π, K ∈ Z is obtained,
The simple decreasing interval is [π / 3 + K π, 5 π / 6 + K π], K ∈ Z,
3. From 2x - π / 6 = π / 2 + K π, K ∈ Z, the symmetry axis is x = π / 3 + (1 / 2) k π, K ∈ Z
4. From 2x - π / 6 = k π, x = π / 12 + (1 / 2) k π, K ∈ Z, the symmetry center is (π / 12 + K π / 2,0), K ∈ Z

The minimum positive period of F (x) = 2Sin (2x - π / 6) + 1 and the symmetry axis and center of image, monotone interval, range of value

The minimum positive period is equal to 2 π / ω is equal to π
The symmetry axis of image x = π / 2K + π / 3
Symmetry center (π / 2K + π / 3,0)
Monotone interval [K π - π / 6, K π + π / 3] increases monotonically [K π + π / 3, K π + 5 π / 12] monotonically decreases
Range [- 1,3]

Find the definition domain of y = - X's square + 2x + 3 and y = x's square-2x-3, when the function value > 0

∵ y = - x ᙽ 2x + 3 ᙽ y = - x ᙽ 2x + 3 ᙽ definition domain: X ∈ R; value range: y ∈ (- ∞, 4]; monotonicity: when x ∈ (- ∞, 1], it increases monotonically; when x ∈ [1, + ∞), it decreases monotonically; axis of symmetry: x = 1

Given the function y = 2Sin (- 2x - π / 4) + 1, find 1. Period 2. The maximum value and the set of X angles when the maximum value is obtained. 3. Monotone interval. 4. Symmetry axis and symmetry center

1 = - 2Sin (2x + π / 4) + 1 = - 2Sin (2x + π / 4) + 1 + T = t = π makes 2x + π / 4 = π / 2 + 2K π (k is an integer) can get x = k π π + 2 + 2K π (k is an integer) can get x = k = k π π + π / 8 (k is an integer) can have the maximum value 3, let 2x + π / 4 = - π / 2 + 2K π (k is an integer) can get x = k π - 3 π / 8 (k is an integer) at this time has the minimum value - 1 Let - 1 make - π / 2 + 2K / 2 + 2K π ≤ 2x + π ≤ 2x + π / π / 4 ≤

Given the function y = 2Sin (2x - π / 4), find the symmetry axis and center

Analysis: the position of the symmetry axis of the image with the function y = 2Sin (2x - π / 4) is the place where the maximum value is taken, and the center of symmetry is the place where the function value is 0
Because 2x - π / 4 = k π + π / 2 (k is an integer), x = k π / 2 + 3 π / 8 is obtained,
So the symmetry axis of the image of the function y = 2Sin (2x - π / 4) is a straight line x = k π / 2 + 3 π / 8 (k is an integer)
Because 2x - π / 4 = k π (k is an integer), x = k π / 2 + π / 8 is obtained,
So the symmetry center of the image of the function y = 2Sin (2x - π / 4) is (K π / 2 + π / 8,0) (k is an integer)

The function y = 2cos (- 2x + π / 3) + 1 is defined in the domain, range, increasing interval, decreasing interval, symmetric axis equation, symmetric center, and so on What

y=2cos( -2x+π/3)+1=2cos( 2x-π/3)+1
Definition field x ∈ R
Range y ∈ [- 1,3]
Increasing interval: K π - π / 3 ≤ x ≤ K π + π / 6, K ∈ Z
Minus interval: K π + π / 6 ≤ x ≤ K π + 2 π / 3, K ∈ Z
Symmetry axis equation x = (K π) / 2 + π / 6
Symmetry center ((K π) / 2 + π / 6,0)

The value range of the function y = 2Sin (3x + 3 / 4 π) is, the monotone increasing interval is, the single point decreasing interval is, the symmetric axis equation is, and the symmetric center coordinate is

The value range of the function y = 2Sin (3x + 3 / 4 π) is [- 2,2], the monotone increasing interval is [2K π / 3-5 π / 12,2k π / 3 - π / 12], (k belongs to Z) monotonic decreasing interval is [2K π / 3 - π / 12,2k π / 3 + π / 4], the symmetric axis equation of (k belongs to Z) is x = (3 + 4K) π / 12, (k

Let f (x) = sin (2x + φ) (- π < φ < 0), y = f (x) be a straight line x = π / 2 1. Find φ 2. Draw the image of function y = f (x) with one period 3. Find the monotone increasing interval of function y = f (x)

1. From F (x) = sin (2x + φ), the symmetry axis is a straight line x = π / 2
When x = π / 2, the function takes the extreme value
Then 2 * π / 2 + φ = k π + π / 2 (K ∈ z)
φ=kπ-π/2
Again - π