Graph three periods of the function. (identify period and phase shift and intercepts)
y = f(x) = √3/2 sin(2x) - 1/2 cos(2x)
if you can't graph it can you at least get it into that simpler form which is graph able.
That can be rewritten
sin(pi/3)sin 2x - cos(pi/3)cos2x
= cos(2x + pi/3)= cos [2(x + pi/6)]
which is a cosine function with an amplitude of 1, a period of pi and a phase shift of -pi/6
To graph the function y = f(x) = √3/2 sin(2x) - 1/2 cos(2x), let's first simplify it into a more graphable form.
We know that sin(2x) = cos(π/2 - 2x). Substituting that into our equation, we get:
y = f(x) = √3/2 cos(π/2 - 2x) - 1/2 cos(2x).
Let's simplify further:
y = f(x) = √3/2 cos(π/2 - 2x) - 1/2 cos(2x)
= √3/2 cos(π/2 - 2x) - 1/2(2cos²(x) - 1) [Using the double angle identity for cosine: cos(2x) = 2cos²(x) - 1]
= √3/2 cos(π/2 - 2x) - cos²(x) + 1/2
Now that we have it simplified, let's analyze the properties of the function.
1. Period:
The period of the function is the distance between two consecutive peaks or troughs of the graph. The period of sin(x) and cos(x) is 2π, but since we have sin(2x) and cos(2x), the period is halved to π. So, our function has a period of π.
2. Phase shift:
The phase shift determines where the graph starts along the x-axis. To find the phase shift, we need to find the value of x that makes the cosine function equal to zero.
Setting cos(x) = 0, we get x = π/2. However, since we have cos(2x), we need to solve for 2x = π/2. Dividing both sides by 2, we get x = π/4. So, the function has a phase shift of π/4 to the right.
3. Intercepts:
To find the y-intercept, we can simply substitute x = 0 into the function:
y = √3/2 cos(π/2 - 2(0)) - cos²(0) + 1/2
= √3/2 cos(π/2) - 1 + 1/2
= √3/2 * 0 - 1 + 1/2
= -1/2.
So, the y-intercept is -1/2.
To find the x-intercepts, we need to solve the equation f(x) = 0:
√3/2 cos(π/2 - 2x) - cos²(x) + 1/2 = 0.
This equation is a bit challenging to solve algebraically. However, we can use a graphing calculator or software to find the x-intercepts. By graphing the function, we can observe where the graph crosses the x-axis.
So, to graph the function and find the x-intercepts, you can use graphing tools or software like Desmos, Mathematica, or even online graphing calculators.