https://file.aiquickdraw.com/mathbot/file/1726151924159u1rwtvst.jpg

Let's address the two problems in the image step by step.

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**Problem 21:**  
The function is given as:
\[ f(x) = -30 \sin(2x) + 15 \]

We need to find the maximum and minimum values of \( f(x) \).

### Solution:
- The sine function \( \sin(2x) \) oscillates between -1 and 1, so the range of \( \sin(2x) \) is:
  \[
  -1 \leq \sin(2x) \leq 1
  \]

- Multiply this range by -30:
  \[
  -30 \times (-1) \leq -30 \sin(2x) \leq -30 \times 1
  \]
  \[
  30 \leq -30 \sin(2x) \leq -30
  \]

- Now add 15 to the entire inequality:
  \[
  30 + 15 \leq -30 \sin(2x) + 15 \leq -30 + 15
  \]
  \[
  45 \leq f(x) \leq -15
  \]

Thus, the **maximum value** is \( 45 \) and the **minimum value** is \( -15 \).

The correct answer is **B) max = 45, min = -15**.

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**Problem 22:**  
We need to identify which of the given statements is true.

Let's examine each option:

**A) The period of \( f(x) = \tan(2\pi x + 1) \) is \( \frac{1}{2} \).**

- The period of the tangent function \( \tan(kx) \) is \( \frac{\pi}{k} \). Here, \( k = 2\pi \), so the period is:
  \[
  \frac{\pi}{2\pi} = \frac{1}{2}
  \]
  This statement is **True**.

**B) The amplitude of \( f(x) = 3 \tan(2x + 3) \) is 3.**

- The tangent function does not have an amplitude because it has vertical asymptotes and ranges from \( -\infty \) to \( \infty \). Therefore, this statement is **False**.

**C) The function \( f(x) = -\frac{1}{2} \sin(2x - 4) + 1 \) is shifted vertically downwards.**

- The function \( f(x) = -\frac{1}{2} \sin(2x - 4) + 1 \) is shifted **vertically upwards** by 1 unit, not downwards. Therefore, this statement is **False**.

**D) The function \( f(x) = 3 \cos(0.5x + 3) + 2 \) is shifted horizontally to the right.**

- The function \( f(x) = 3 \cos(0.5x + 3) + 2 \) has a phase shift. To find the phase shift, solve \( 0.5x + 3 = 0 \):
  \[
  x = -\frac{6}{0.5} = -6
  \]
  This indicates a shift to the **left** by 6 units, not to the right. Therefore, this statement is **False**.

Thus, the correct answer is **A) The period of \( f(x) = \tan(2\pi x + 1) \) is \( \frac{1}{2} \)**.

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Would you like more details or have any other questions? Here are some related questions for practice:

1. How do you find the period of a sine or cosine function?
2. What is the range of a general sine function \( f(x) = A \sin(Bx) + C \)?
3. How does the amplitude of a cosine function affect its graph?
4. What are the key transformations for the graph of a tangent function?
5. Can you describe how horizontal phase shifts affect the graph of trigonometric functions?

**Tip:** Always remember that the amplitude only applies to sine and cosine functions, while tangent and cotangent have no amplitude.

Solve these problems: If the function f(x) = -30 sin(2x) + 15, what are the maximum and minimum values of f(x)? Which of the following statement is true?

Learn how to find the maximum and minimum values of the function f(x) = -30 sin(2x) + 15 and identify true statements about trigonometric function properties. This problem covers key concepts in trigonometry, function analysis, and period determination, suitable for students in Grades 10-12.

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