Y = a cos(bx + c) + d

Definition

This equation represents the general form of a cosine function, where 'a' indicates the amplitude, 'b' affects the period, 'c' is the phase shift, and 'd' represents the vertical shift. Understanding this equation helps in graphing trigonometric functions and recognizing their key characteristics such as maximum and minimum values, symmetry, and periodic behavior.

5 Must Know Facts For Your Next Test

1. The amplitude 'a' indicates how far the graph reaches above and below its midline; a larger value results in taller waves.
2. The period of the cosine function is influenced by 'b'; a higher value of 'b' leads to a shorter wavelength and faster oscillation.
3. The phase shift 'c' adjusts the starting position of the wave along the x-axis, allowing for horizontal translations of the graph.
4. The vertical shift 'd' moves the entire graph up or down without changing its shape, affecting where it oscillates relative to the x-axis.
5. Graphs of cosine functions are even functions, which means they are symmetric about the y-axis, a key feature to note when analyzing their behavior.

Review Questions

• How does changing the value of 'a' in the equation y = a cos(bx + c) + d affect the graph of the cosine function?
  • Changing 'a' alters the amplitude of the cosine function. A larger absolute value of 'a' results in taller peaks and deeper troughs, meaning that the graph stretches vertically. Conversely, a smaller absolute value compresses the graph towards the midline. This change affects how far above and below the midline (y = d) the graph extends.
• What is the effect of adjusting 'b' on both the period and frequency of the cosine function represented by y = a cos(bx + c) + d?
  • Adjusting 'b' modifies both the period and frequency of the cosine wave. The period is calculated as $$\frac{2\pi}{b}$$; thus, increasing 'b' shortens the period, resulting in more cycles occurring within a given interval on the x-axis. This means that as 'b' increases, frequency also increases since frequency is inversely related to period.
• Evaluate how a combination of changes to 'c' and 'd' can influence both position and overall shape of y = a cos(bx + c) + d when graphed.
  • When both 'c' and 'd' are adjusted simultaneously, 'c' shifts the graph horizontally while 'd' shifts it vertically. For instance, if you increase 'c', it moves the graph left or right depending on its sign, thus altering where one complete cycle starts. Increasing 'd' raises or lowers this entire wave on the y-axis. Together, these adjustments maintain the same amplitude and period but change where on the Cartesian plane this wave oscillates.

Related terms

Amplitude:

The amplitude is the maximum distance from the midline of the wave to its peak or trough, calculated as the absolute value of 'a' in the cosine function.

Period:

The period is the distance along the x-axis required for the function to complete one full cycle, determined by the formula $$\frac{2\pi}{b}$$.

Phase Shift:

The phase shift refers to the horizontal displacement of the graph of a trigonometric function, which can be determined by $$-\frac{c}{b}$$.