graph y=-3sec(2x-pi/3)proved the period vertical translation and the phase shift
write it as y = -3sec 2(x-π/6)
period = 2π/2 = π
no vertical translation
phase shift π/6 units to the right
To understand the given equation and determine its period, vertical translation, and phase shift, let's break it down step by step.
First, let's examine the general form of the equation: y = asec(bx + c) + d
In this form, 'a' represents the amplitude, 'b' represents the horizontal compression/stretch, and 'c' represents the horizontal shift. The constant 'd' represents the vertical shift.
Now, let's compare this with the given equation: y = -3sec(2x - π/3)
From this, we can deduce the following information:
1. Amplitude (a): In the given equation, there is no coefficient directly multiplying the secant function. Therefore, the amplitude is undefined, indicating that there is no vertical stretching or compression.
2. Horizontal compression/stretch (b): The coefficient preceding 'x' is 2. This value determines the horizontal compression/stretch. In this case, b = 2, which means the graph is compressed horizontally compared to the standard secant function.
3. Horizontal shift (c): The value being subtracted inside the secant function, 2x - π/3, indicates the horizontal shift. By setting this expression equal to zero and solving, we can find the value of 'x' that corresponds to the phase shift.
2x - π/3 = 0
2x = π/3
x = π/6
Therefore, the graph experiences a phase shift of π/6 units to the right.
4. Vertical shift (d): The constant term -3 indicates the vertical translation of the graph. In this case, the graph is shifted down by 3 units.
Now, let's summarize the findings:
- Period: The period of the secant function is 2π divided by the absolute value of 'b.' In this case, since b = 2, the period is 2π/2 = π.
- Vertical translation: The graph is shifted downward by 3 units due to the vertical shift (d = -3).
- Phase shift: The graph is shifted to the right by π/6 units due to the horizontal shift (c = π/6).
In conclusion, the equation y = -3sec(2x - π/3) represents a secant function with a period of π, a vertical shift of -3 units, and a phase shift of π/6 units to the right.