The range of $ \sqrt{13} \sin(x + \phi) $ is $ [-\sqrt{13}, \sqrt{13}] $.

["Understanding the Range of ( \sqrt{13} \sin(x + \phi) ): A Complete Guide", "When analyzing periodic functions in trigonometry, one essential concept is the range of a sine-based expression. In this article, we explore the range of the expression ( \sqrt{13} \sin(x + \phi) ), where ( \phi ) is a phase shift. Whether you're a student, educator, or self-learner, understanding this concept is crucial for mastering trigonometric functions and their applications.", "---", "### What is ( \sqrt{13} \sin(x + \phi) )?", "The function ( \sqrt{13} \sin(x + \phi) ) is a transformed sine wave. Here:", "- ( \sqrt{13} ) is the amplitude, which controls the vertical stretch of the wave.\n- ( x + \phi ) represents a phase shift, moving the function horizontally.\n- ( \phi ) is a constant phase shift—positive shifts the graph left, negative shifts it right—but does not affect the range.", "Since sine functions naturally oscillate between -1 and 1, multiplying by ( \sqrt{13} \ scales this interval vertically.", "---", "### The Mathematical Derivation", "We start from the fundamental identity:\n[\n-1 \leq \sin(x + \phi) \leq 1\n]", "Multiplying the entire inequality by ( \sqrt{13} > 0 ), the direction of the inequality remains unchanged:\n[\n- \sqrt{13} \leq \sqrt{13} \sin(x + \phi) \leq \sqrt{13}\n]", "Thus, the range of ( \sqrt{13} \sin(x + \phi) ) is clearly:\n[\n[-\sqrt{13}, \sqrt{13}]\n]", "---", "### Why the Range Matters", "Understanding the range is essential for several reasons:", "- Graphing Accuracy: Knowing the vertical bounds helps plot the function correctly on a coordinate plane.\n- Real-World Applications: In physics and engineering, sine waves model oscillating phenomena—knowing their range ensures valid predictions.\n- Problem Solving: Many trigonometric equations and inequalities depend on identifying minimum and maximum values.", "---", "### Visualizing the Range: Graph Insights", "If you graph ( y = \sqrt{13} \sin(x + \phi) ), it repeats every ( 2\pi ) units (due to the sine function’s period), and every peak and trough hits exactly ( \pm \sqrt{13} ). The phase shift ( \phi ) moves the entire wave left or right but does not stretch or compress it vertically.", "---", "### Practical Example", "Suppose you model a sound wave or alternating voltage using:\n[\ny = \sqrt{13} \sin(2x + \pi/4) + 5\n]\nHere, the amplitude ( \sqrt{13} ) stretches the wave, and the phase shift ( \pi/4 ) shifts it. However, the full range remains:\n[\n5 - \sqrt{13} \leq y \leq 5 + \sqrt{13}\n]\nWhich simplifies to approximately ( 5 - 3.6 = 1.4 ) to ( 5 + 3.6 = 8.6 ), still fitting ( [-\sqrt{13}, \sqrt{13}] ) vertically after adjusting the baseline at ( y = 5 ).", "---", "### Final Thoughts", "The expression ( \sqrt{13} \sin(x + \phi) ) reaches all values between ( -\sqrt{13} ) and ( \sqrt{13} ), bounded tightly by its amplitude and completely unaffected by phase shifts. Recognizing this range empowers you to analyze and apply trigonometric functions confidently across math, physics, and engineering.", "Key Takeaway: For any function of the form ( A \sin(x + \phi) ), the range is always ( [-|A|, |A|] ). In this case, ( A = \sqrt{13} ), so the range is exactly:", "[\n[-\sqrt{13}, \sqrt{13}]\n]", "---", "Mastering trigonometric ranges like this unlocks deeper insight into periodic behavior and enhances problem-solving precision. Keep practicing, and soon you’ll see sine waves—and their limits—clearly and confidently."]