Y = -2cosx
The equation \( y = -2 \cos x \) represents a cosine function that has been transformed in a few ways:
Amplitude: The amplitude of the cosine function is given by the coefficient in front of the cosine. In this case, the amplitude is \( |-2| = 2 \), which means the graph will oscillate between -2 and 2.
Vertical Shift: There is no vertical shift in this function since there is no added constant term.
Reflection: The negative sign in front of the 2 indicates that the graph of the cosine function is reflected over the x-axis.
Period: The period of the standard cosine function is \( 2\pi \). Since there are no additional horizontal transformations (like a change in frequency), the period remains \( 2\pi \).
Key Points of the Graph
- Maximum value: The maximum value of \( y \) is 0 (when \( \cos x = 0 \)).
- Minimum value: The minimum value is -2 (when \( \cos x = 1 \)).
- Intercepts:
- To find the x-intercepts, set \( y = 0 \): \[ 0 = -2 \cos x \implies \cos x = 0 \] This occurs at \( x = \frac{\pi}{2} + k\pi \) for \( k \in \mathbb{Z} \).
Summary
The graph of \( y = -2 \cos x \) is a cosine wave that oscillates between 0 and -2, has a period of \( 2\pi \), and is reflected across the x-axis.
