Answer:
The attached graph is the answer
Step-by-step explanation:
* Lets revise the trigonometry translation
- If the equation is y = A sec(B(x + C)) + D
# Amplitude is A
- The Amplitude is the height from the center line to the peak .
Or we can measure the height from highest to lowest points and
divide that by 2
# period is 2π/B
- The Period goes from one peak to the next
# phase shift is C (positive is to the left)
- The Phase Shift is how far the function is shifted horizontally
from the usual position.
# vertical shift is D
- The Vertical Shift is how far the function is shifted vertically from
the usual position.
- If y = sec(x)
∴ A = 1 , B = 1 , C = 0 , D = 0
- That means the amplitude is 1, the period is 2π, no phase shift
or vertical shift
* Now lets solve the problem
∵ y = 0.5 sex(x + π/3) - 2
∴ A = 0.5
- The 0.5 tells us it will be half times taller than usual,
so Amplitude = 0.5
∴ B = 1
- The period is 2π
∴ C = π/3
- The π/3 means it will be shifted to the left by π/3
∴ D = -2
- The -2 tells us the vertical Shift = -2
* Lets calculate the y-intercept
- To find the y-intercept put x = 0
∴ y = 0.5 sec(0 + π/3) - 2
∵ sec(π/3) = 2
∴ y = 0.5(2) - 2 = 1 - 2 = -1
* Now lets look to the graphs to find the correct answer
- Only one graph has y-intercept is -1
- Look to the attached graph
- There is a vertical shift down 2 units
- There is a horizontal shift to the left by π/3 units
- The amplitude is 1/2 ⇒ (-1.5 - -2.5)/2 = 1/2, where -1.5 and -2.5
are the peak points
- The y-intercept is -1
* The attached graph is the answer
- For more understand look to the attached color graph
- The blue graph is y = sec(x)
- The red graph is y = 0.5 sec(x + π/3) - 2
