How do you graph #h(x) = 2 sin(πx−3π) − 4#?

Question

Oliver Atkinson · Accepted Answer

As below.

#h(x) = 2 sin (pi x - 3pi) - 4# Standard form is #y = A sin (Bx - C) + D# #A = 2, B = pi, C = 3pi, D = -4# #Amplitude = |A| = 2# #"Period " = (2pi) / |B| = (2pi) / pi = 2# #"Phase Shift " = -C / B = (3pi) / pi = 3#, #color(red)(3 " to the right"# #"Vertical Shift " = D = -4# graph{2 sin (pi x - 3pi) - 4 [-10, 10, -5, 5]}

Elijah Alexander · Answer

undefined To graph the function $ h(x) = 2 \sin(\pi x - 3\pi) - 4 $, follow these steps:

1. Identify the key components of the function:
   - Amplitude: $ 2 $
   - Period: $ \frac{2\pi}{\pi} = 2 $ (since the coefficient of $ x $ in $ \sin(\pi x - 3\pi) $ is $ 1 $, the period is $ \frac{2\pi}{1} = 2\pi $ and the frequency is $ \frac{1}{2\pi} $)
   - Phase shift: $ \frac{3\pi}{\pi} = 3 $ (positive indicates a shift to the right)

2. Plot the key points based on the standard sine function:
   - The midline is $ y = -4 $.
   - Key points for one period of the sine function are at $ \left(0, -4 + 2\right) = (0, -2) $, $ \left(\frac{1}{2}, -4\right) $, $ \left(1, -4 - 2\right) = (1, -6) $, and so on.

3. Apply the phase shift by shifting the key points to the right by $ 3 $ units.

4. Sketch the graph by connecting the points smoothly, maintaining the shape of the sine function.

5. Label the axes and any important points on the graph.

Following these steps will help you accurately graph the function $ h(x) = 2 \sin(\pi x - 3\pi) - 4 $.