which of the following statements about $f(x) = \sin\left(\frac{\pi}{4}x\
ight)$ is true?

- the graph of $f$ is increasing on $0 \le x \le 4$ and also on $16 \le x \le 20$ because these intervals are separated by a length of exactly one period.
- the graph of $f$ is increasing on $0 \le x \le 4$ and also on $16 \le x \le 20$ because these intervals are separated by a length of exactly $2\pi$ units.
- the graph of $f$ is increasing on $-2 \le x \le 2$ and also on $6 \le x \le 10$ because these intervals are separated by a length of exactly one period.
- the graph of $f$ is increasing on $-2 \le x \le 2$ and also on $6 \le x \le 10$ because these intervals are separated by a length of exactly $2\pi$ units.

Question

which of the following statements about $f(x) = \sin\left(\frac{\pi}{4}x\
ight)$ is true?

- the graph of $f$ is increasing on $0 \le x \le 4$ and also on $16 \le x \le 20$ because these intervals are separated by a length of exactly one period.
- the graph of $f$ is increasing on $0 \le x \le 4$ and also on $16 \le x \le 20$ because these intervals are separated by a length of exactly $2\pi$ units.
- the graph of $f$ is increasing on $-2 \le x \le 2$ and also on $6 \le x \le 10$ because these intervals are separated by a length of exactly one period.
- the graph of $f$ is increasing on $-2 \le x \le 2$ and also on $6 \le x \le 10$ because these intervals are separated by a length of exactly $2\pi$ units.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Sinusoidal Period Analysis", "Trigonometric Monotonicity" ], "new_concepts": [], "current_concepts": [ "Sinusoidal Period Analysis", "Trigonometric Monotonicity", "Periodic Functions" ] } ## Find the period of the function Using the *Sinusoidal Period Analysis* knowledge point $$ T = \frac{2\pi}{B} = \frac{2\pi}{\frac{\pi}{4}} = 8 $$ ## Determine the interval of increase Using the *Trigonometric Monotonicity* knowledge point $$ \begin{aligned} &-\frac{\pi}{2} \le \frac{\pi}{4}x \le \frac{\pi}{2} \ &-2 \le x \le 2 \end{aligned} $$ ## Apply periodicity to find other intervals Using the *Sinusoidal Period Analysis* knowledge point $$ \begin{aligned} &[-2 + T, 2 + T] = [-2 + 8, 2 + 8] = [6, 10]\ &\text{Separation: } 6 - (-2) = 8 = 1\text{ period} \end{aligned} $$

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

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

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

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

{ "elements": [ { "type": "functiongraph", "params": [ { "js": "Math.sin((Math.PI/4)*x)", "latex": "\sin(\frac{\pi}{4}x)" }, -4, 12 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "f(x)", "withLabel": true } }, { "type": "segment", "params": [ [-2, -1], [2, 1] ], "properties": { "strokeColor": "#F2557F", "strokeWidth": 4, "name": "Increasing Interval 1" } }, { "type": "segment", "params": [ [6, -1], [10, 1] ], "properties": { "strokeColor": "#F2557F", "strokeWidth": 4, "name": "Increasing Interval 2" } } ], "timestamps": [0.5] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Sinusoidal Period Analysis" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Find the period of the function

Determine the interval of increase

Apply periodicity to find other intervals

Answer: