Problem 3 - Precalculus

Question

For the equation \(A \cos (B x+C)+D,\) what constants affect the range of the function and how affect the range?

Step-by-Step Solution

Answer

Constants \(A\) and \(D\) affect the range; \(A\) changes magnitude and \(D\) shifts it.

Step 1: Understand the Function's General Form

The given function is of the form \[A \cos (B x + C) + D\].This is a transformed cosine function where:- \(A\) affects the amplitude,- \(B\) affects the period,- \(C\) is a phase shift,- \(D\) is a vertical shift.

Step 2: Calculate the Basic Range of Cosine Function

The basic cosine function \(\cos(x)\) varies between -1 and 1. Therefore, its range is \([-1, 1]\).

Step 3: Determine the Effect of Amplitude (A) on Range

The amplitude \(A\) scales the cosine function vertically. The range of the transformed function \(A \cos(Bx+C)\) is \([-A, A]\). This means \(A\) stretches (if \(|A|>1\)) or compresses (if \(0<|A|<1\)) the range vertically.

Step 4: Determine the Effect of Vertical Shift (D) on Range

The constant \(D\) shifts the entire function vertically. Thus, it adds \(D\) units to both the minimum and the maximum value of the cosine wave. Therefore, the complete range of the function \(A \cos(Bx+C) + D\) is \([-A+D, A+D]\).

Key Concepts

Amplitude

Amplitude refers to the height of the peaks and the depth of the troughs in a trigonometric function. In the context of the function \( A \cos(Bx+C) + D \), the amplitude is represented by the constant \( A \). The amplitude directly affects how far the function's values reach above and below the central axis. For a standard cosine function, the values oscillate between -1 and 1. However, when amplitude \( A \) is introduced, this range is altered to span from \(-A\) to \(A\). The amplitude essentially scales the height of the cosine wave:

• If \(|A| > 1\), the wave stretches vertically, making peaks higher and troughs lower.
• If \(0 < |A| < 1\), the wave compresses, reducing the distance between the peaks and troughs and the central axis.

This scaling effect is crucial for determining how dramatically the function oscillates and is important in applications where the maximum and minimum values of the function reflect physical constraints, such as waves.

Vertical Shift

The vertical shift of a trigonometric function refers to how the function moves up or down on a graph. In the function \( A \cos(Bx+C) + D \), the vertical shift is expressed by the constant \( D \). The role of the vertical shift is to adjust the midline of the graph from the horizontal axis. Here's how it affects the function:

• Adding \( D \) to the function lifts every point on the graph upwards by \( D \) units.
• Subtracting or using a negative \( D \) lowers every point on the graph by \( D \) units.

The effect of the vertical shift is seen in the adjusted range of the function. Originally, a cos function ranges between \(-A\) and \(A\), but after incorporating a vertical shift, it becomes \([-A+D, A+D]\). This shift is significant because it influences where the maximum and minimum points of the function sit relative to the horizontal axis, crucial for modeling phenomena like tides, sound waves, and alternating current heights.

Range of Function

The range of a function describes all the possible output values it can have. For trigonometric functions like \( A \cos(Bx+C) + D \), understanding the range is key to predicting the function's behavior. Initially, the range of the basic cosine function \( \cos(x) \) is from -1 to 1. However, the presence of amplitude \( A \) and vertical shift \( D \) transforms this range significantly:

• The amplitude \( A \) means the function now spans from \(-A\) to \(A\).
• The vertical shift \( D \) further modifies the range, resulting in \([-A+D, A+D]\).

This transformed range is critical for interpreting the function's full behavior. In practical applications, knowing the full scope of possible values helps in designing systems that use these trigonometric functions, such as engineering signals or predicting natural cycles. The combination of amplitude and vertical shift ensures that the range is both scalable and adjustable for different contexts.