Problem 38 - Algebra 2

Question

Write an equation for each translation. \(y=\sin x, 2\) units to the right and 4 units down

Step-by-Step Solution

Answer

The equation for the sine function after a shift of 2 units to the right and four units down is \(y = \sin (x-2) - 4\).

Step 1: Identify the original function and the shifts

The original function is \(y=\sin x\). It is to be shifted 2 units to the right and 4 units down.

Step 2: Apply horizontal shift

For a horizontal (left or right) shift, the form of the equation is \(y = \sin (x-h)\), where h is the magnitude of the horizontal shift. Subtract h if the shift is to the right, or add h if the shift is to the left. Consequentially, our new equation after the shift of 2 units to the right becomes \(y = \sin (x-2)\).

Step 3: Apply vertical shift

For a vertical (up or down) shift, the form of the equation is \(y = \sin x + k\), where k is the magnitude of the vertical shift. Add k if the shift is upwards, and subtract k if the shift is downwards. Here, the function shifts 4 units down, so our final transformed function becomes \(y = \sin (x-2) - 4\).

Key Concepts

Horizontal Shifts

Horizontal shifts are movements of a function along the x-axis. If you have a function like \( y = \sin x \), a horizontal shift affects the phase of the function. When shifting horizontally:

• Moving to the right involves subtracting a value from the x-variable, represented as \( x - h \), where \( h \) is the number of units you move to the right.
• Conversely, moving to the left means adding to the x-variable, as in \( x + h \).

For example, if you shift the sine function 2 units to the right, the equation becomes \( y = \sin(x-2) \). The graph will appear as if it's been slid smoothly along the x-axis without any vertical alteration. This change in the equation affects the starting position of one cycle of the sine wave along the horizontal axis, which is crucial for syncing the wave with real-world phenomena.

Vertical Shifts

Vertical shifts modify the graph of a function by moving it up or down along the y-axis. Such shifts don't alter the pattern of the wave itself but rather reposition the entire wave vertically:

• Shifting upwards involves adding a number, resulting in the equation \( y = \sin x + k \), where \( k \) is the units moved up.
• Downwards shifts are represented by subtracting a number, as in \( y = \sin x - k \).

In our exercise, shifting the sine function 4 units down results in the equation \( y = \sin (x-2) - 4 \). This shifts every point on the sine wave down by 4 units, affecting the amplitude line but keeping the wave's shape intact. Each peak, trough, and the midpoint of the wave is equally moved downward, making it lower in the vertical plane.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are fundamental to mathematics, especially in modeling periodic phenomena such as sound waves, tides, and circular motion. These functions have specific properties that make them unique and wildly useful:

• Periodic Nature: Sine and cosine functions repeat their values in regular intervals, known as periods. This characteristic is crucial in applications like electronics and engineering.
• Wave Patterns: The patterns produced by trigonometric functions are smooth and wave-like, with consistent frequency and amplitude.
• Standard Form: The basic form \( y = \sin x \) can be adjusted using parameters to shift, stretch, or compress the wave.

Understanding these functions allows for comprehensive analysis and transformation, which is essential for harmonizing or synchronizing these waves with given systems or problems. In transformations, leveraging the natural structure of these functions provides elegance and simplicity to complex motions and patterns.