Problem 19 - Algebra 2

Question

Describe any phase shift and vertical shift in the graph. $$ y=\sin \left(x+\frac{\pi}{2}\right)+2 $$

Step-by-Step Solution

Answer

The function has a phase shift of \(-\frac{\pi}{2}\) to the left and a vertical shift of '2' upward.

Step 1: Identify the Phase Shift

In the function \(y=\sin \left(x+\frac{\pi}{2}\right)+2\), the \(x+\frac{\pi}{2}\) inside the sine function represents the phase shift. The graph of the sine function is shifted to the left by \(\frac{\pi}{2}\) units. The phase shift is \(-\frac{\pi}{2}\).

Step 2: Identify the Vertical Shift

The '+2' that is added to the sine function represents the vertical shift. This shifts the graph of the function up by 2 units. Thus, the vertical shift is '2'.

Key Concepts

Phase Shift

In trigonometry, phase shift refers to the horizontal translation of a periodic function, such as the sine function.
When analyzing the equation \(y=\sin\left(x+\frac{\pi}{2}\right)\), the component \(x+\frac{\pi}{2}\) indicates a phase shift.
The term "phase shift" tells us how far the entire graph moves along the x-axis.

In this equation, the graph is shifted to the left by \(\frac{\pi}{2}\) units.
This is because adding \(\frac{\pi}{2}\) inside the function means you start the wave earlier in its cycle.

• A positive phase shift makes the graph move to the left.
• A negative phase shift makes it move to the right.

Understanding phase shifts helps in graphing and predicting trigonometric functions more accurately.
In simpler terms, a phase shift changes where the wave starts on the graph.

Vertical Shift

The vertical shift is a transformation that moves a graph up or down on the coordinate plane.
It's represented by adding or subtracting a number outside the sine function.
In \(y=\sin\left(x+\frac{\pi}{2}\right)+2\), the number "+2" shows a vertical shift.

This means the entire graph moves up by 2 units.
It affects the middle or baseline of the sine wave, raising all the points by two on the y-axis.

• If the added number is positive, the graph shifts upward.
• If the number is negative, the graph shifts downward.

Such shifts are straightforward yet crucial, as they influence the position of a graph relative to the x-axis, but do not alter its shape.

Sine Function

The sine function is one of the primary trigonometric functions that models wave-like patterns.
It's represented mathematically as \(y = \sin(x)\).
This function oscillates between 1 and -1, creating a smooth curve known as a sine wave.

Features of the sine function:

• Period: The function repeats every \(2\pi\) units.
• Amplitude: The height from the middle to the peak, typically 1 for the standard sine function.
• Frequency: How often the wave repeats within a given interval.

The sine function is often utilized in physics and engineering to model phenomena such as sound waves, light waves, and other cyclical patterns.
Understanding it is integral to analyzing any oscillatory or periodic data.

Graph Transformations

Graph transformations modify the appearance of graphs by shifting, stretching, or flipping them.
A graph can be changed in aspect but maintains its fundamental shape.

Types of Transformations:

• Translations: Moving the graph horizontally or vertically without altering its shape.
• Stretches: Making the graph wider or narrower.
• Reflections: Flipping the graph across an axis.

In the context of the sine function, transformations such as phase and vertical shifts are key tools.
They help describe the changes in the function's position and orientation.
The combined effect of these transformations allows the graph to be manipulated to meet specific criteria without losing its identity as a sine wave.