Question

URGENTWhich of the following is the graph of y=0.5sec (x+pi/3)-2?

Answer 1

Answer:

See attachment.

Step-by-step explanation:

The given function is

$y=0.5\sec(x+\frac{\pi}{3})-2$

This is the graph of $y=\sec(x)$ that is shifted down 2 units and shifted to the left $\frac{\pi}{3}$ units.

When x=0, $y=0.5\sec(0+\frac{\pi}{3})-2$

$y=0.5\sec(\frac{\pi}{3})-2$

$y=1-2=-1$

The y-intercept is -1.

The correct choice is shown in the attachment

Answer 2

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