To analyze the effect of the coefficient [tex]\(-2\)[/tex] on the function [tex]\( y = -2 \cos (x - \pi) \)[/tex], let's break down the components of the transformation one by one.
1. Amplitude Change:
- The coefficient [tex]\( -2 \)[/tex] affects the amplitude of the cosine function. The basic cosine function, [tex]\( \cos(x) \)[/tex], has an amplitude of 1.
- When a cosine function is multiplied by a negative coefficient, the amplitude changes to the absolute value of that coefficient, which is 2 in this case. So the amplitude is now 2.
- A positive coefficient would just stretch or compress the graph vertically depending on its value. Since the coefficient is -2, it not only stretches by a factor of 2 but also reflects the graph across the horizontal axis.
2. Vertical Stretch and Reflection:
- The factor of 2 specifically means the graph is vertically stretched by a factor of 2. This means all y-values are multiplied by 2.
- The sign (negative) in front of 2 indicates a reflection over the horizontal axis. This means each point on the graph is flipped over the x-axis.
Summarizing these observations:
- The original [tex]\(\cos(x)\)[/tex] function is vertically stretched by a factor of 2.
- Additionally, the graph is reflected over the horizontal axis due to the negative sign.
Thus, the correct effect of the [tex]\(-2\)[/tex] coefficient on the function [tex]\(\cos(x - \pi)\)[/tex] is a vertical stretch by a factor of 2 and then a flip over the horizontal axis.
Therefore, the answer is:
- C. Vertical stretch by factor 2 then flip over horizontal axis.
