The vector is $ \boxed{\begin{bmatrix} 7 \\ -7 \end{bmatrix}} $.Question: A volcanologist models the amplitude of seismic waves using the function $ f(x) = \sin^2 x + 2\cos x $. Find the range of $ f(x) $ as $ x $ ranges over all real numbers.

["Exploring the Range of $ f(x) = \sin^2 x + 2\cos x $: A Volcanological Modeling Insight", "Seismic activity remains one of Earth’s most dynamic and complex phenomena. As a volcanologist modeling subsurface wave propagation, understanding the behavior of physical functions like $ f(x) = \sin^2 x + 2\cos x $ is essential to interpreting seismic data. This article combines mathematics and geophysics to analyze the range of this function—revealing the amplitude limits of modeled seismic wave components.", "### Understanding the Function", "We are given:\n$$\nf(x) = \sin^2 x + 2\cos x\n$$\nOur goal is to determine the full range of $ f(x) $ for all real $ x $. To do this, we rewrite the trigonometric terms using identities, reducing the expression to a single-variable function easier to analyze.", "Recall the Pythagorean identity:\n$$\n\sin^2 x = 1 - \cos^2 x\n$$\nSubstituting:\n$$\nf(x) = (1 - \cos^2 x) + 2\cos x = -\cos^2 x + 2\cos x + 1\n$$\nLet $ u = \cos x $. Since the cosine function satisfies $ -1 \leq \cos x \leq 1 $, we restrict $ u \in [-1, 1] $. Then:\n$$\nf(u) = -u^2 + 2u + 1, \quad u \in [-1, 1]\n$$", "### Analyzing the Quadratic Function", "The expression $ f(u) = -u^2 + 2u + 1 $ is a quadratic in standard form $ f(u) = -(u^2 - 2u - 1) $. The coefficient of $ u^2 $ is negative, so the parabola opens downward, meaning it has a maximum at its vertex.", "The vertex occurs at:\n$$\nu = -\frac{b}{2a} = -\frac{2}{2(-1)} = 1\n$$\nSince $ u = 1 $ lies within $ [-1, 1] $, we evaluate $ f(u) $ at $ u = 1 $:\n$$\nf(1) = -(1)^2 + 2(1) + 1 = -1 + 2 + 1 = 2\n$$\nThis is the maximum value.", "Now evaluate at the other endpoint $ u = -1 $:\n$$\nf(-1) = -(-1)^2 + 2(-1) + 1 = -1 - 2 + 1 = -2\n$$", "### Determining the Range", "Because $ f(u) $ is continuous on $ [-1, 1] $, and the parabola peaks at $ u = 1 $ with value 2 and decreases to $ f(-1) = -2 $, the function attains all values from its minimum to its maximum over this interval.", "Thus, the range of $ f(u) $ on $ [-1, 1] $ is $ [-2, 2] $. Therefore, the range of $ f(x) = \sin^2 x + 2\cos x $ is:\n$$\n\boxed{[-2, 2]}\n$$", "### Real-World Implication", "For the volcanologist, this bounded output range confirms that modeled seismic wave amplitudes fluctuate within predictable limits. Understanding this has direct applications in distinguishing background noise from potential volcanic tremors—key in early hazard detection.", "### Conclusion", "By transforming $ \sin^2 x + 2\cos x $ into a quadratic in $ \cos x $, we efficiently determined its range using domain restrictions and calculus. The result $[-2, 2]$ not only satisfies mathematical rigor but also supports practical geophysical modeling. As seismic data grow more complex, tools like this empower scientists to decode Earth’s rhythmic pulsing beneath the surface.", "---", "Let $ \mathbf{v} = \boxed{\begin{bmatrix} 7 \ -7 \end{bmatrix}} $.\nThis vector, while not directly tied to the amplitude function, symbolizes directional coherence in multidimensional seismic data—reminding us that even simple components combine to reveal Earth’s inner voice.", "Keywords: $ f(x) = \sin^2 x + 2\cos x $, range, cosine function, seismic wave modeling, volcanology, trigonometric functions, precalculus, amplitude analysis, $ [-2, 2] $, mathematical modeling."]