To determine if the given function [tex]\( y = 2 \sec \left( x + \frac{\pi}{6} \right) + 2 \)[/tex] matches with one of the provided options, we should break down the components of the function step by step.
1. Identify the base trigonometric function:
The function involves the secant function, denoted as [tex]\( \sec \)[/tex].
2. Understand the transformation inside the secant function:
The expression inside the secant function is [tex]\( x + \frac{\pi}{6} \)[/tex]. This indicates a horizontal shift of [tex]\( -\frac{\pi}{6} \)[/tex]. Shifts are due to the addition of the phase constant [tex]\( \frac{\pi}{6} \)[/tex] to the variable [tex]\( x \)[/tex].
3. Scaling factor (vertical stretch/compression):
The coefficient [tex]\( 2 \)[/tex] in front of the secant function means that the secant function is stretched vertically by a factor of 2.
4. Vertical translation:
The constant [tex]\( +2 \)[/tex] added outside the secant function signifies a vertical shift upwards by 2 units.
Putting these observations together, the given function [tex]\( y = 2 \sec \left( x + \frac{\pi}{6} \right) + 2 \)[/tex] can be understood as:
- a base [tex]\( \sec \)[/tex] function,
- horizontally shifted [tex]\( \frac{\pi}{6} \)[/tex] units to the left,
- vertically stretched by a factor of 2,
- and translated upwards by 2 units.
Therefore, the equation of the function is:
[tex]\[ y = 2 \sec \left( x + \frac{\pi}{6} \right) + 2 \][/tex]
This breakdown supports that the given function is correctly represented as [tex]\( y = 2 \sec \left( x + \frac{\pi}{6} \right) + 2 \)[/tex].
