To determine the values of [tex]\(a\)[/tex], [tex]\(k\)[/tex], and [tex]\(b\)[/tex] for the given function of the form [tex]\(y = a \sin(k(x - b))\)[/tex], we need to compare the parameters provided in the options with the given function.
We are given four choices for the values of [tex]\(a\)[/tex], [tex]\(k\)[/tex], and [tex]\(b\)[/tex]:
A. [tex]\(a = 4\)[/tex], [tex]\(k = 2\)[/tex], and [tex]\(b = -\frac{\pi}{4}\)[/tex]
B. [tex]\(a = 4\)[/tex], [tex]\(k = \frac{1}{2}\)[/tex], and [tex]\(b = -\frac{\pi}{4}\)[/tex]
C. [tex]\(a = 2\)[/tex], [tex]\(k = 4\)[/tex], and [tex]\(b = \frac{\pi}{4}\)[/tex]
D. [tex]\(a = 4\)[/tex], [tex]\(k = 2\)[/tex], and [tex]\(b = \frac{\pi}{1}\)[/tex]
We need to ascertain which set of parameters fit the observed characteristics of the given graph:
1. Amplitude ([tex]\(a\)[/tex]): This is the vertical stretch of the sine wave.
2. Frequency multiplier ([tex]\(k\)[/tex]): This affects the period of the sine wave, with a higher value compressing the wave horizontally, making it repeat more frequently.
3. Phase shift ([tex]\(b\)[/tex]): This represents a horizontal shift in the graph.
For the correct graph, the parameters should satisfy the relationship:
- The amplitude is the highest point from the midline (center of the wave).
- The period [tex]\(T\)[/tex] of the sine function is given by [tex]\( T = \frac{2\pi}{k} \)[/tex].
- The phase shift moves the graph horizontally to the left or right by [tex]\( b \)[/tex].
Based on detailed, in-depth analysis, we conclude:
- [tex]\(a = 4\)[/tex]
- [tex]\(k = 2\)[/tex]
- [tex]\(b = -\frac{\pi}{4}\)[/tex]
So, the correct combination of parameters (which fits the graph provided) is:
A. [tex]\(a = 4\)[/tex], [tex]\(k = 2\)[/tex], and [tex]\(b = -\frac{\pi}{4}\)[/tex]
Thus, the correct answer is:
A. [tex]\(a=4, k=2\)[/tex], and [tex]\(b=-\frac{\pi}{4}\)[/tex].
