Problem 40 - Algebra 2

Question

Write an equation for each translation. \(y=\sin x, \frac{\pi}{2}\) units to the right and 3.5 units up

Step-by-Step Solution

Answer

The translated function is \(y=\sin(x-\frac{\pi}{2})+3.5\)

Step 1 : Translate Horizontally

The effect of shifting sinx horizontally \(\frac{\pi}{2}\) units to the right will be to replace x with \(x-\frac{\pi}{2}\) So, the intermediate function after performing this translation is \(y=\sin(x-\frac{\pi}{2})\).

Step 2: Translate Vertically

Moving any function 3.5 units upwards means adding 3.5 to the function. So, we add 3.5 to the function from Step 1: \(y=\sin(x-\frac{\pi}{2})+3.5\). This is our final function.

Key Concepts

Horizontal Translation

A horizontal translation refers to shifting a function right or left along the x-axis. When we talk about the sine function, this means altering its standard sine wave position. To shift a function horizontally, we adjust the input variable (generally \(x\)). For instance, if we want to shift \(y = \sin x\) to the right by \(\frac{\pi}{2}\) units, we replace \(x\) with \(x - \frac{\pi}{2}\). The result of this operation gives us the new intermediate function: \(y = \sin(x - \frac{\pi}{2})\). This signifies the wave has moved to the right by \(\frac{\pi}{2}\) units. ul>- Replace \(x\) with \(x - \text{shift distance}\) to move right- Replace \(x\) with \(x + \text{shift distance}\) to move left
Understanding this translation helps in visualizing how the sine curve shifts along the horizontal axis while maintaining its shape. This concept is widely applicable in trigonometric functions and real-world contexts such as signal processing or wave modeling.

Vertical Translation

Vertical translation involves shifting a function up or down along the y-axis. For sine functions, this refers specifically to how much the amplitude or baseline shifts vertically. To carry out a vertical shift, we simply add or subtract a constant from the function.

In our example, we are asked to shift the sine function \(3.5\) units upward. Starting from our horizontally shifted function \(y = \sin(x - \frac{\pi}{2})\), we add \(3.5\) to obtain the final equation: \(y = \sin(x - \frac{\pi}{2}) + 3.5\). This modification effectively raises the entire wave up by \(3.5\) units without altering its frequency or period.

- Add a positive constant to shift upwards- Subtract a positive constant to shift downwards

This transformation is useful in real-life scenarios such as adjusting sine waves to reflect changes in baseline levels, such as rising tides or electricity levels.

Sine Function

The sine function is a fundamental mathematical function that occurs frequently in trigonometry, represented by \(\sin x\). This function depicts a periodic wave that cycles smoothly between -1 and 1 over a cycle of \(2\pi\), known as a period.

Key characteristics of the sine function include its regular amplitude, which is the peak value (1 or -1 for the basic sine function), and its period, which represents the length of one full cycle of the wave from start to finish. The graph of the sine function starts at the origin, rises to a peak, returns to the middle, falls to a trough, and then returns to the origin again, thereby completing one cycle.

- Amplitude: Maximum height (from baseline to peak or trough)- Period: Length of one complete cycle; \(2\pi\) for \(\sin x\)- Midline: The central axis around which the wave oscillates

The sine function's regularity and predictability make it invaluable in various fields—and understanding how it transforms through horizontal and vertical translations is essential for modeling oscillatory behavior in physics and engineering.