Let's go through the transformations step by step to understand which statements about the function [tex]\( h(x) = -3 \cos(2x - \pi) + 4 \)[/tex] are correct.
1. Phase Shift (Horizontal Shift):
The phase shift can be found by examining the horizontal shift in the argument of the cosine function [tex]\(2x - \pi\)[/tex]. We set [tex]\(2x - \pi = 0\)[/tex] and solve for [tex]\(x\)[/tex]:
[tex]\[
2x - \pi = 0 \implies x = \frac{\pi}{2}
\][/tex]
So, the phase shift is [tex]\(\frac{\pi}{2}\)[/tex] units to the left. This statement is correct.
2. Vertical Compression:
The coefficient in front of the cosine function, [tex]\(-3\)[/tex], indicates a vertical compression by a factor of 3 and a reflection across the x-axis. This statement is partially correct, but it only mentions the compression and omits the reflection. So, this statement needs more details to be fully correct.
3. Vertical Shift:
The "+4" term at the end of the function indicates a vertical shift upwards by 4 units. This statement is correct.
4. Period Change:
The period of the parent cosine function [tex]\(\cos(x)\)[/tex] is [tex]\(2\pi\)[/tex]. The period of [tex]\( \cos(2x) \)[/tex] is obtained by dividing the parent period by the coefficient of [tex]\(x\)[/tex]:
[tex]\[
\text{New Period} = \frac{2\pi}{2} = \pi
\][/tex]
So, the period of the function [tex]\(h(x)\)[/tex] is [tex]\(\pi\)[/tex], which is half the period of the parent function. This statement is correct.
5. Amplitude Change:
The amplitude of the parent cosine function [tex]\(\cos(x)\)[/tex] is 1. When multiplied by [tex]\(-3\)[/tex], the amplitude becomes [tex]\(3\)[/tex] (amplitude is always positive regardless of the sign). This indicates the stretching factor. This statement is correct if we interpret the compression factor as the absolute value for the amplitude, which the correct conversion process does imply.
The correct statements in Hillary's description about the function [tex]\(h(x)\)[/tex] are:
- Phase shift left [tex]\(\frac{\pi}{2}\)[/tex] units.
- The graph of the parent function is vertically compressed by a factor of 3 (including a reflection if we consider negative sign).
- The graph of the parent function is vertically shifted up by 4 units.
- The period of the function [tex]\(h\)[/tex] is [tex]\(\pi\)[/tex], which is half the period of the parent function.
- The amplitude of the new function is 3.
In summary, the correct statements featured in the provided answer are all valid, emphasizing the correct transformations of the parent function to obtain [tex]\(h(x)\)[/tex].
