Sinusoidal functions

Trig functions like sine and cosine have periodic graphs which we called Sinusoidal Graph, or Sine wave.

Sinusoidal graphs

Graph of unit sine & cosine function:

Midline, amplitude and period

They're three features of sinusoidal graphs.

• Midline: is the horizontal line that passes exactly in the middle between the graph's maximum and minimum points.
• Amplitude: is the vertical distance between the midline and one of the extremum points.
• Period: Also called frequency, is the distance between two consecutive maximum points, or two consecutive minimum points (these distances must be equal).

Initial period and how to graph a sinusoidal function

To graph the whole function, you only need 1 period of the graph, and then just repeat that ever and ever.
And to avoid any confusion, we'd pick the initial period, which:

• sin(x) that interval is [0,2π], and looks like an S.
• cos(x) that interval is [0,2π], and looks like a Bow.

You really need to pay attention for the starting point of the S and the bow, it's the great way to figure out the how much the graph shifted on x-axis.

The necessary information to draw the initial period :

• Any two of the three position: Max, Min and Midline
• Period
• Phase shift from initial point

With these information above, we could figure out all other informations about the period.

Initial period of sin(x):

Looks like a flipped S.

Starting point of the flipped S is its Midpoint, at (0,0),
Full period is 2π, Midline at y=0, Range is [1,-1].

Initial period of cos(x):

Looks like a bow.

Starting point of the bow is its Extreme point, at (0,1),
Full period is 2π, Midline at y=0, Range is [1,-1].

Phase shift of trig functions

Phase shift means horizontal shift, or moves on x-axis. It's much harder to understand and calculate than the vertical shift.

Since trig functions(sine, cosine, tangent) are all periodic functions, so it's really CONFUSING with horizontal moving, because it's repeating, and you can't easily tell what happened with the graph.

Calculate algebraically how much is the Phase Shift

First need to figure out the starting point of the initial period, and to compare how much it moved from 0 on x-axis.

Since the initial period of both sine and cosine functions starts from 0 on x-axis,
with the formula of function y = A*sin(Bx+C)+D,

we are to set the (Bx+c) = 0, and solve for x,

the value of x is the phase shift of the graph.

Why do we set (Bx+c) = 0?
Because we could imagine the (Bx+c) is a whole, and could be w of the sin(w).
Since the initial period of sin(w), always starts from 0,
we could say the starting point of initial period is 0, so w=0, then Bx+C=0.

Informations of Sinusoidal functions

For the General sinusoidal function:

f(x) = A・sin(Bx + C) + D

• Amplitude: |A|
• Period: |2π / B|
• Vertical shift: D
• Midline: y=D
• Range: [-A+D, A+D]
• Phase shift: Set (Bx+C)=0, and solve for x.
• Sign of sine: ~+ if there's a Max point after intersect y-axis, - if Minimum point.~

Example:

f(x) = - 2sin(2x + 3) + 10

• Amplitude: |-2| = 2
• Period: |2π / 2| = π
• Vertical shift: +10
• Midline: y=10
• Range: [2+10, -2+10] = [12, 8]
• Horizontal shift: set 2x+3=0, get x=-3/2, so it's shifting -3/2 from origin.

Example 2:

Solve:

• Midline: y=2.5
• Amplitude: |6-2.5|=3.5
• Period: Midpoint to Max is 1/4 period, so 1/4 * 2π/B = |-4π - (-5/2)π|, solve for B gets 1/3.
• Phase shift: -4π. Because the midpoint on the left is the starting point of initial period

At this moment, our formula is almost finished:
y = 3.5*sin(1/3 *x +C) +2.5, so only the C is not yet solved.
set 1/3 *x + C = 0, since , since -4π is the phase shift, so we set x=-4π, solve C gets 4π/3.