Problem 19 - Algebra 2 and Trigonometry

Question

Find the phase shift of each function. \(y=\cos \left(x+\frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

The phase shift is \(-\frac{\pi}{2}\), meaning the graph shifts left by \(\frac{\pi}{2}\) units.

Step 1: Identify the General Form

The general form of the function is given by \(y = \cos(Bx + C)\)." This formula allows us to identify the phase shift.

Step 2: Match with the Given Function

Our given function is \(y = \cos(x + \frac{\pi}{2})\)." By comparing this with the general form, we identify \(B = 1\) and \(C = \frac{\pi}{2}\)."

Step 3: Calculate the Phase Shift

The phase shift is calculated using the formula \(\text{Phase Shift} = -\frac{C}{B}\)." Substituting the values found in Step 2, we get \(-\frac{\frac{\pi}{2}}{1} = -\frac{\pi}{2}\)."

Step 4: Determine the Direction of the Shift

A negative phase shift indicates the function shifts to the left. Therefore, the phase shift here is \(-\frac{\pi}{2}\)," which means the graph shifts left by \(\frac{\pi}{2}\) units."

Key Concepts

Cosine Function

The cosine function, denoted as \( y = \cos(x) \), is one of the basic trigonometric functions used to model periodic phenomena. Originating from the unit circle, the cosine function represents the x-coordinate of a point as it rotates around the circle. This function starts at a maximum value of 1 when \( x = 0 \) and reaches its minimum value of -1 at odd multiples of \( \pi \).Common characteristics of the cosine function include:

• Periodicity: The cosine function repeats every \(2\pi\) radians, which is its period.
• Range: Between -1 and 1.
• Even Function: The cosine function is even, meaning \( \cos(-x) = \cos(x) \).
• Symmetry: Symmetric around the y-axis.

When a cosine function is modified by adding constants, its graph experiences shifts and stretches, altering its position and shape without changing its basic properties like periodicity or range.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, are essential in mathematics for describing relationships in triangles and modeling periodic behaviors. These functions come from the study of angles and the unit circle, where a right triangle's hypotenuse acts as the radius.Some key trigonometric functions include:

• Cosine (\(y = \cos(x)\)): Dictates the horizontal aspect, representing how far a point is along the x-axis on the unit circle.
• Sine (\(y = \sin(x)\)): Refers to the vertical component, showcasing how far a point is along the y-axis.
• Tangent (\(y = \tan(x)\)): A combination of sine and cosine, calculated as \( \frac{\sin(x)}{\cos(x)} \), representing the slope.

Trigonometric functions are utilized widely in various disciplines such as physics, engineering, and computer science, primarily for solving problems involving oscillations, waves, and other cyclic patterns.

Graph Transformations

Graph transformations involve altering the appearance and position of a function's graph. Understanding these transformations is crucial in comprehending how different algebraic changes affect the visual representation of a graph.Key graph transformations include:

• Vertical and Horizontal Shifts: Adding or subtracting a constant to the function shifts its graph up or down (vertical) or left or right (horizontal). For example, \( y = \cos(x + \frac{\pi}{2}) \) represents a horizontal shift.
• Reflections: Multiplying the function by a negative value reflects it over the x-axis or y-axis.
• Stretching and Compressing: Altering the coefficient in front of \( x \) changes the period and amplitude of the function. Stretching makes the graph wider, while compressing makes it narrower.

For the specific example of \( y = \cos(x + \frac{\pi}{2}) \), the graph undergoes a phase shift to the left by \( \frac{\pi}{2} \) units due to the negative sign in the phase shift calculation, transforming the position of the original cosine wave.