Trigonometric functions can be shifted horizontally and vertically, changing their behavior. Phase shifts move the graph left or right, affecting when events occur. Vertical shifts move the entire graph up or down, altering the baseline value.

These shifts are crucial in modeling real-world phenomena. By understanding how to apply and interpret them, you can create more accurate representations of periodic events like temperature fluctuations, sound waves, or planetary orbits.

Understanding Shifts in Trigonometric Functions

Phase and vertical shifts

• Phase shift horizontally moves graph measured in x-axis units left (positive) or right (negative) (π/2 left)
• Vertical shift moves graph up or down y-axis units (3 units up)
• Effects on function: phase shift alters input (x) while vertical shift changes output (y)
• Phase shift impacts timing of periodic events (sunrise 1 hour earlier)
• Vertical shift adjusts baseline or average value of function (temperature fluctuations around higher average)

Shifts in function equations

• General form: $f(x) = A \sin(B(x - C)) + D$ or $f(x) = A \cos(B(x - C)) + D$
• Phase shift (C) value subtracted from x inside parentheses, C units right
• Vertical shift (D) constant term added outside trigonometric function, D units up
• Example: $f(x) = \sin(x - \frac{\pi}{4}) + 2$ has π/4 right phase shift and 2 units up vertical shift
• Negative shifts: $g(x) = \cos(x + \frac{\pi}{3}) - 1$ moves π/3 left and 1 unit down

Graphing shifted trigonometric functions

• Steps: start with parent function, apply phase shift, then vertical shift
• Plot key points: maxima, minima, x-intercepts, y-intercept
• Verify graph crosses midline correctly, has proper amplitude and period
• Example: $y = \sin(x - \frac{\pi}{2}) + 1$
  1. Shift $\sin(x)$ right by π/2
  2. Move entire graph up 1 unit
  3. Check y-intercept now at (0, 1) instead of (0, 0)

Combining shifts for complex functions

• Amplitude (A) affects graph height, stretches/compresses vertically
• Period ($\frac{2\pi}{B}$) changes graph width, stretches/compresses horizontally
• Apply transformations: period, phase shift, amplitude, vertical shift
• Interpret complex functions by analyzing each transformation separately
• Example: $f(x) = 2\sin(\frac{\pi}{3}(x + 1)) - 4$
  1. Period: $\frac{2\pi}{B} = \frac{2\pi}{\frac{\pi}{3}} = 6$
  2. Phase shift: 1 unit left
  3. Amplitude: 2
  4. Vertical shift: 4 units down