Problem 45 - Algebra 2

Question

OPEN ENDED Write the equation of a trigonometric function with a phase shift of \(-45^{\circ} .\) Then graph the function, and its parent graph.

Step-by-Step Solution

Answer

The equation is \(y = \sin(x + 45^{\circ})\). Shift the sine graph 45° left.

Step 1: Understand Phase Shift

A phase shift in the equation of a trigonometric function means the graph of the function is shifted horizontally. A negative phase shift indicates a shift to the right. Here, the phase shift is \(-45^{\circ}\).

Step 2: Choose a Trigonometric Function

Let's choose the basic sine function, \(y = \sin(x)\), to create a function with the desired phase shift.

Step 3: Write the Function with Phase Shift

The general form of a sine function with a phase shift is \(y = \sin(x - \phi)\), where \(\phi\) is the phase shift. For a phase shift of \(-45^{\circ}\), the function becomes \(y = \sin(x + 45^{\circ})\).

Step 4: Prepare the Parent Function Graph

Graph the parent function \(y = \sin(x)\). This function has a period of \(360^{\circ}\), with key points at \(0^{\circ}, 90^{\circ}, 180^{\circ}, 270^{\circ},\) and \(360^{\circ}\).

Step 5: Graph the Function with Phase Shift

Graph \(y = \sin(x + 45^{\circ})\). This shifts the parent sine function \(45^{\circ}\) to the left, starting the cycle at \(-45^{\circ}\). The shape of the sine wave remains unchanged, just the start and end points have shifted.

Key Concepts

Phase Shift

When working with trigonometric functions, a phase shift refers to the horizontal movement of a graph along the x-axis. This shift changes where the cycle of the function starts. A positive phase shift moves the graph to the left, while a negative one moves it to the right.

To determine the phase shift in the function, we use the expression inside the sine function's parentheses. For example, in the expression \(y = \sin(x + \phi)\), the \(\phi\) value represents the phase shift. If \(\phi\) is \(-45^{\circ}\), it reflects a shift of 45 degrees to the left. This is crucial for graphing, as it alters where key points like peaks and troughs appear. By understanding phase shifts, one can predict and plot trigonometric graphs more accurately.

Sine Function

The sine function is a fundamental trigonometric function that describes a smooth, periodic oscillation. It is typically written as \(y = \sin(x)\). Key characteristics of the sine function include:

• Its period, or the length of one complete cycle, is \(360^{\circ}\).
• Ithas an amplitude (height) of 1, translating to peak values of 1 and minimum values of -1.
• The nature of the graph results in a smooth, wave-like curve.

The sine function starts at the point \(x = 0^{\circ}\) and progresses through key points at \(90^{\circ}, 180^{\circ}, 270^{\circ},\) and returning to \(360^{\circ}\). Understanding these properties helps in visualizing and manipulating the sine graph. When phase shifts or amplitude changes are introduced, these points will adjust according to the specified transformation.

Graphing Trigonometric Functions

Graphing trigonometric functions requires capturing their key properties such as the amplitude, period, and phase shift.

To effectively graph these functions, follow these steps:

• Identify the parent function, such as \(y = \sin(x)\) or \(y = \cos(x)\).
• Determine any transformations, including phase shifts, amplitude changes, and vertical or horizontal dilations.
• Mark key points on the x-axis that correspond to one period of the function, adjusting for any phase shift.
• Plot the corresponding y-values considering amplitude changes and trace the curve across the period.

Graphing involves systematically applying transformations to the parent graph and ensuring the shape retains its sine or cosine pattern.

Parent Graph

The parent graph of a trigonometric function is the simplest form from which different transformations are applied. For sine functions, this means starting with \(y = \sin(x)\).

The parent sine function graph looks like a continuous wave oscillating between -1 and 1 through the cycle of 0 to \(360^{\circ}\). Key points are:

• Starts at \(0, 0\)
• Maximum at \(90^{\circ}, 1\)
• Back to 0 at \(180^{\circ}\)
• Minimum at \(270^{\circ}, -1\)
• Completes at \(360^{\circ}, 0\)

Such a graph serves as the base to understand any phase shift, amplitude change, or period adjustment, providing a framework for predicting how transformations will alter the overall graph.