Problem 96 - Algebra and Trigonometry

Question

Describe the relationship between the graphs of \(y=A \cos (B x-C)\) and \(y=A \cos (B x-C)+D\)

Step-by-Step Solution

Answer

The addition of D in the function \(y=A \cos (B x-C)+D\) versus \(y=A \cos (B x-C)\) introduces a vertical shift. For positive D, the graph moves up by D units, for negative D, it moves down by |D| units. Other characteristics like amplitude, period and phase shift remain unchanged.

Step 1: Understand the General Form

The general form of a cosine function is given by \(y=A \cos (B x-C)\) where A is the amplitude, B influences the period, and C corresponds to the phase shift.

Step 2: Understand the Function \(y=A \cos (B x-C)\)

The graph of \(y=A \cos (B x-C)\) is a wave that oscillates between -A and A. It has a period of \(\frac{2 \pi}{B}\) and is horizontally shifted C units to the right.

Step 3: Understand the Effect of Adding D

The term D introduces a vertical shift in the graph. If D > 0, the graph \(y=A \cos (B x-C)+D\) will shift D units upwards from \(y=A \cos (B x-C)\). If D < 0, the graph will shift |D| units downwards.

Step 4: Understand the Function \(y=A \cos (B x-C)+D\)

The graph of \(y=A \cos (B x-C)+D\) has the same shape as \(y=A \cos (B x-C)\) but is vertically shifted D units. It oscillates between -A+D and A+D, has a period of \(\frac{2 \pi}{B}\), and is horizontally shifted C units to the right.