Which of the following statements about f ( x )

To analyze which statement is true, let's carefully study the function $f(x) = \sin\left(\frac{\pi x}{4}\right)$:

Step 1: Identify the period of $f(x)$

The function $f(x) = \sin\left(\frac{\pi x}{4}\right)$ has a period of:

$T = \frac{2\pi}{\frac{\pi}{4}} = 8$

This means the sine wave repeats every 8 units along the $x$-axis.

Step 2: Determine the intervals where $f(x)$ is increasing

The sine function increases from: $0 \text{ to } \pi \quad \text{(i.e., from the start to the middle of one period)}.$

For $\sin\left(\frac{\pi x}{4}\right)$, this occurs on the interval where: [ \frac{\pi x}{4} \in [0, \pi] \implies x \in [0, 4] ]

This pattern will repeat for any interval of length 8 because the period is 8. For example:

• The graph of $f$ is increasing on $x \in [0, 4]$ and $x \in [16, 20]$, since these intervals are separated by a full period $(T = 8)$.

Step 3: Check each statement

1. The graph of $f$ is increasing on $0 \leq x \leq 4$ and also on $16 \leq x \leq 20$ because these intervals are separated by a length of exactly one period.

   • True: These intervals are separated by one period (8 units), and the graph is increasing on both.

2. The graph of $f$ is increasing on $0 \leq x \leq 4$ and also on $16 \leq x \leq 20$ because these intervals are separated by a length of exactly $2\pi$ units.

   • False: $2\pi \approx 6.28$, which is not the period of this function.

3. The graph of $f$ is increasing on $-2 \leq x \leq 2$ and also on $6 \leq x \leq 10$ because these intervals are separated by a length of exactly one period.

   • False: $-2 \leq x \leq 2$ is not one of the increasing intervals, as the function increases on $[0, 4]$ and repeats every 8 units.

4. The graph of $f$ is increasing on $-2 \leq x \leq 2$ and also on $6 \leq x \leq 10$ because these intervals are separated by a length of exactly $2\pi$ units.

   • False: The reasoning is incorrect for the same reasons as above.

Correct Answer:

The graph of $f$ is increasing on $0 \leq x \leq 4$ and also on $16 \leq x \leq 20$ because these intervals are separated by a length of exactly one period.

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Do you want further clarification or visualization? Here are 5 questions to expand on this topic:

1. What is the importance of the period in identifying increasing/decreasing intervals?
2. How do we generalize the intervals for increasing behavior of periodic functions?
3. What would happen if the coefficient inside the sine function changed?
4. Can you explain why $2\pi$ is irrelevant in this case?
5. How do transformations affect the period and phase of a sine function?

Tip: Always calculate the period first for trigonometric functions to identify repetitive patterns in behavior.