1.
Item 2. The range of the trigonometric function \(y=a\cos x+b\) is \(-2\leq y\leq 8\). Determine the value of \(b\).
For a sinusoidal function, \(b\) represents the vertical translation, also called the equilibrium line or midline. The midline is the average of the maximum and minimum values. \(b=\frac{\text{max}+\text{min}}{2}\) The maximum value is \(8\), and the minimum value is \(-2\). \(b=\frac{8+(-2)}{2}\) \(b=\frac{6}{2}=3\) Therefore \(b=3\). Answer: C
2.
Item 4. If the period of the function \(y=\sin(bx)\) is \(\frac{3\pi}{5}\), find \(b\).
For \(y=\sin(bx)\), the period is \(\frac{2\pi}{b}\). The given period is \(\frac{3\pi}{5}\). Use \(b=\frac{2\pi}{\text{period}}\). \(b=\frac{2\pi}{\frac{3\pi}{5}}\) \(b=2\pi\cdot\frac{5}{3\pi}\) \(b=\frac{10}{3}\) Therefore \(b=\frac{10}{3}\). Answer: A
3.
Item 3. Of the following equations, which have undergone a horizontal expansion from their original equations \(y=\sin x\) and \(y=\cos x\)? I. \(y=2\cos\left(\frac{3}{4}x\right)\) II. \(y=-\sin\left(\frac{3}{2}x\right)\) III. \(y=3\cos\left(\frac{5}{4}x\right)\) IV. \(y=5\sin\left(\frac{4}{3}x\right)\) V. \(y=\frac{1}{2}\cos\left(\frac{1}{3}x\right)\)
For \(y=\sin(bx)\) or \(y=\cos(bx)\), the horizontal scale factor is \(\frac{1}{|b|}\). A horizontal expansion happens when \(0<|b|<1\). I. \(y=2\cos\left(\frac{3}{4}x\right)\): here \(b=\frac{3}{4}\), so \(0<|b|1\). This is a horizontal contraction. III. \(y=3\cos\left(\frac{5}{4}x\right)\): here \(b=\frac{5}{4}>1\). This is a horizontal contraction. IV. \(y=5\sin\left(\frac{4}{3}x\right)\): here \(b=\frac{4}{3}>1\). This is a horizontal contraction. V. \(y=\frac{1}{2}\cos\left(\frac{1}{3}x\right)\): here \(b=\frac{1}{3}\), so \(0<|b|<1\). This is a horizontal expansion. Therefore the equations with a horizontal expansion are I and V. Answer: D
4.
Item 5. A mass is supported by a spring so that it rests \(30\text{ cm}\) above a table top. The mass is pulled down to a height of \(10\text{ cm}\) above the table top and released at time \(t=0\). It takes \(0.8\) seconds for the mass to reach a maximum height of \(50\text{ cm}\) above the table top. As the mass moves up and down, its height \(h\), above the table top, is approximated by a sinusoidal function of the elapsed time \(t\), for a short period of time. Determine which of the following is the cosine equation that gives the height, \(h\), as a function of time, \(t\).
The minimum height is \(10\text{ cm}\), and the maximum height is \(50\text{ cm}\). The amplitude is \(\frac{\text{max}-\text{min}}{2}\). \(\text{Amplitude}=\frac{50-10}{2}=20\) The vertical displacement is \(\frac{\text{max}+\text{min}}{2}\). \(d=\frac{50+10}{2}=30\) From the minimum height to the maximum height takes \(0.8\) seconds, which is half of the full period. \(\text{Period}=2(0.8)=1.6\) For a cosine model, \(b=\frac{2\pi}{\text{period}}=\frac{2\pi}{1.6}\). Since the mass starts at the minimum height when \(t=0\), use a negative cosine model. Therefore the equation is \(h=-20\cos\left(\frac{2\pi t}{1.6}\right)+30\). Answer: D
5.
Item 6. What is the period of the function \(y=2\tan\left(\frac{x}{2}+3\right)\)?
The period of \(y=\tan(bx)\) is \(\frac{\pi}{|b|}\). First identify the coefficient of \(x\). \(y=2\tan\left(\frac{x}{2}+3\right)\) \(y=2\tan\left(\frac{1}{2}x+3\right)\), so \(b=\frac{1}{2}\). \(\text{Period}=\frac{\pi}{\frac{1}{2}}\) \(\text{Period}=2\pi\) Therefore the period is \(2\pi\). Answer: D
6.
Item 1. Determine the horizontal translation of the function \(y=2\cos(3x-12)\).
To determine the horizontal translation, first factor the coefficient of \(x\) inside the cosine function. \(y=2\cos(3x-12)\) \(y=2\cos\left(3(x-4)\right)\) The expression is now in the form \(y=a\cos(b(x-c))+d\). Here \(c=4\), so the graph is translated \(4\) units to the right. Answer: D
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