Notes on Translating Periodic Functions

Translating Periodic Functions

Vertical Translation

• Periodic functions like sine and cosine can be translated, stretched, or shrunk.
• Vertical translation shifts the entire function up or down.
• To determine the vertical translation, identify the midline of the translated function.
• The midline is the horizontal line that runs midway between the maximum and minimum values of the function.
• Example:
  • A cosine function, which normally oscillates around the x-axis (midline at $y = 0$), is shifted upwards.
  • The maximum value of the translated function is 5, and the minimum value is 3.
  • The midline is at $y = 4$, which is the average of 3 and 5: $(\frac{3+5}{2} = 4)$.
• To translate the graph 4 units up, replace $y$ with $(y - 4)$.
• The equation becomes $y - 4 = \cos(x)$, or equivalently, $y = \cos(x) + 4$.

Horizontal Translation

• Horizontal translation shifts the function to the left or right.
• Example:
  • The sine function, $y = \sin(x)$, has key points at $(0, 0)$, $(\frac{\pi}{2}, 1)$, $(\pi, 0)$, $(\frac{3\pi}{2}, -1)$, and $(2\pi, 0)$.
  • If the sine function is translated $\frac{\pi}{2}$ units to the right, all the x-coordinates are increased by $\frac{\pi}{2}$.
  • The new starting point is at $(\frac{\pi}{2}, 0)$.
  • The translated graph completes one full period (2\pi) at $x = \frac{5\pi}{2}$.
• To shift the graph $\frac{\pi}{2}$ units to the right, replace $x$ with $(x - \frac{\pi}{2})$.
• The equation becomes $y = \sin(x - \frac{\pi}{2})$.

Combined Translations

• A function can be translated both vertically and horizontally.
• To determine the equation of a translated function, identify how far the original function has been shifted in both the x and y directions.
• Example:
  • The sine function is translated so that its starting point is at $(2, 3)$.
  • This means the graph has been shifted 2 units to the right and 3 units up.
• Replace $x$ with $(x - 2)$ to represent the horizontal shift, and replace $y$ with $(y - 3)$ to represent the vertical shift.
• The equation becomes $y - 3 = \sin(x - 2)$, or equivalently, $y = \sin(x - 2) + 3$.

Real-World Applications

• Data from experiments involving periodic phenomena (e.g., sound waves, light waves, seasonal changes) can be modeled using translated sine or cosine functions.
• By determining the period, amplitude, and translations from the data, one can find a function that fits the data.

Calculator Demonstration

• A calculator can be used to visualize the effects of horizontal and vertical translations on a sine function.
• The general form of the function is $y = \sin(x - c) + d$, where $c$ represents the horizontal translation and $d$ represents the vertical translation.
• Changing the value of $c$ slides the graph horizontally: negative $c$ shifts it to the right, and positive $c$ shifts it to the left.
• Changing the value of $d$ slides the graph vertically: increasing $d$ shifts it up, and decreasing $d$ shifts it down.

General Equation

• By combining horizontal/vertical stretches/shrinks and horizontal/vertical translations, one can fit a sine or cosine function to a wide range of periodic data.
  • Horizontal stretch or shrink adjusts the period.
  • Vertical stretch or shrink adjusts the amplitude.
  • Horizontal translation shifts the function horizontally.
  • Vertical translation shifts the function vertically.