To determine the midline, amplitude, and range of the function [tex]\( g(x) = -\frac{1}{2} \cos (4x) + 1 \)[/tex], let's break down each part of the function and analyze it step-by-step.
### Midline
The midline of the function is determined by the constant term added or subtracted from the trigonometric function. In the given function [tex]\( g(x) = -\frac{1}{2} \cos (4x) + 1 \)[/tex], the constant term is 1. This constant term represents the vertical shift of the cosine function and dictates the midline.
Midline: [tex]\( y = 1 \)[/tex]
### Amplitude
The amplitude of a cosine function is determined by the coefficient in front of the cosine term. The amplitude is the absolute value of this coefficient. For [tex]\( g(x) = -\frac{1}{2} \cos (4x) + 1 \)[/tex], the coefficient in front of the cosine term is -[tex]\(\frac{1}{2}\)[/tex]. The absolute value of this coefficient is [tex]\(\frac{1}{2}\)[/tex].
Amplitude: [tex]\( \frac{1}{2} \)[/tex]
### Range
The range of the function is influenced by both the amplitude and the midline. The amplitude affects how much the function oscillates around the midline, while the midline determines the center of these oscillations.
The highest value of the function occurs when the cosine term is at its maximum value of 1:
[tex]\[ g(x) = -\frac{1}{2} \cdot 1 + 1 = -\frac{1}{2} + 1 = \frac{1.5}{} \][/tex]
[tex]\[ Highest\ value = 1 + \frac{1}{2} = 1.5 \][/tex]
The lowest value of the function occurs when the cosine term is at its minimum value of -1:
[tex]\[ g(x) = -\frac{1}{2} \cdot (-1) + 1 = \frac{1}{2} + 1 = 1.5{} \][/tex]
[tex]\[ Lowest\ value = 1 - \frac{1}{2} = 0.5 \][/tex]
Therefore, the range of the function is from [tex]\( 0.5 \)[/tex] to [tex]\( 1.5 \)[/tex].
Range: [tex]\(\left[ 0.5, 1.5 \right]\)[/tex]
### Summary
Based on the analysis:
- The midline is [tex]\( y = 1 \)[/tex].
- The amplitude is [tex]\( \frac{1}{2} \)[/tex].
- The range is [tex]\(\left[0.5, 1.5 \right]\)[/tex].
Here are the pairs you need to match:
- Midline: [tex]\( y = 1 \)[/tex]
- Amplitude: [tex]\( \frac{1}{2} \)[/tex]
- Range: [tex]\(\left[0.5, 1.5 \right]\)[/tex]
This completes the detailed step-by-step solution for determining the midline, amplitude, and range of the function [tex]\( g(x) = -\frac{1}{2} \cos (4x) + 1 \)[/tex].
