### Comparing Functions: A Detailed Analysis---##### Compare the following functions:- **Function \( f(x) \)**: \[ f(x) = -4 \sin(3x - \pi) + 2 \] This function represents a sinusoidal wave modified by a sine term with amplitude of -4, a frequency coefficient of 3, and a phase shift of \(\pi\), shifted vertically by 2 units.- **Function \( g(x) \)**: - Graph Description: The provided graph of \( g(x) \) appears to be a parabola opening upwards. The vertex of the parabola looks to be at the lowest point on the graph. - Critical Points (approx): - Vertex: The vertex, which appears to be the minimum point, is at approximately \(x = 3\) and \(y = -5\).- **Function \( h(x) \)**: - Table of Values: \[ \begin{array}{|c|c|} \hline x & y \\ \hline -2 & 14 \\ \hline -1 & 9 \\ \hline 0 & 6 \\ \hline 1 & 5 \\ \hline 2 & 6 \\ \hline 3 & 9 \\ \hline 4 & 14 \\ \hline \end{array} \] From the table, we see that \( h(x) \) has its minimum value at \( x = 1 \) and \( y = 5 \).---##### Question: Which function has the smallest minimum?- \( f(x) \)- \( g(x) \)- \( h(x) \)- All three functions have the same minimumBy analyzing the functions:- \( f(x) \) will have a minimum based on its periodic nature and the amplitude of the sine function, which can be calculated to be at \( y = -4 + 2 = -2 \).- \( g(x) \) obtains a minimum at around \( y = -5 \), as seen from the vertex in the graph.- \( h(x) \) has its minimum at \( y =

Answered: Image /qna-images/answer/b7aad819-a8f3-4ab6-854b-51bf21006173.jpg

Compare the following functions: g(x) |f(x) = -4 sin(3x − π) + 2 Which function has the smallest minimum? Of(x) O g(x) Oh(x) O All three functions have the same minimum P h(x) y -2 14 -19 0 6 1 5 2 6 3 9 4 14

Compare the following functions: g(x) |f(x) = -4 sin(3x − π) + 2 Which function has the smallest minimum? Of(x) O g(x) Oh(x) O All three functions have the same minimum P h(x) y -2 14 -19 0 6 1 5 2 6 3 9 4 14

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

### Comparing Functions: A Detailed Analysis

---

##### Compare the following functions:

- **Function \( f(x) \)**:
\[
f(x) = -4 \sin(3x - \pi) + 2
\]
This function represents a sinusoidal wave modified by a sine term with amplitude of -4, a frequency coefficient of 3, and a phase shift of \(\pi\), shifted vertically by 2 units.

- **Function \( g(x) \)**:
- Graph Description:
The provided graph of \( g(x) \) appears to be a parabola opening upwards. The vertex of the parabola looks to be at the lowest point on the graph.

- Critical Points (approx):
- Vertex: The vertex, which appears to be the minimum point, is at approximately \(x = 3\) and \(y = -5\).

- **Function \( h(x) \)**:
- Table of Values:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & 14 \\
\hline
-1 & 9 \\
\hline
0 & 6 \\
\hline
1 & 5 \\
\hline
2 & 6 \\
\hline
3 & 9 \\
\hline
4 & 14 \\
\hline
\end{array}
\]

From the table, we see that \( h(x) \) has its minimum value at \( x = 1 \) and \( y = 5 \).

---

##### Question: Which function has the smallest minimum?
- \( f(x) \)
- \( g(x) \)
- \( h(x) \)
- All three functions have the same minimum

By analyzing the functions:

- \( f(x) \) will have a minimum based on its periodic nature and the amplitude of the sine function, which can be calculated to be at \( y = -4 + 2 = -2 \).
- \( g(x) \) obtains a minimum at around \( y = -5 \), as seen from the vertex in the graph.
- \( h(x) \) has its minimum at \( y =

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