Sinusoidal Function Transformations - AP Precalculus

1. In the function $g(x) = A\cos(Bx+C)+D$, which parameter must be changed to alter the period of the function?
   Answer: $B$, which controls the horizontal dilation. The period of the function $g(x) = A\cos(Bx+C)+D$ is given by $P = \frac{2\pi}{|B|}$. Therefore, changing the value of $B$ alters the period by causing a horizontal stretch or compression.
2. Which of the following describes the transformations applied to the graph of $y = \sin(x)$ to obtain the graph of $g(x) = \sin(x + \frac{\pi}{3}) - 5$?
   Answer: A horizontal shift of $\frac{\pi}{3}$ units to the left and a vertical shift of 5 units down. In the general form $y = a\sin(b(x-c))+d$, $c$ represents the horizontal shift and $d$ represents the vertical shift. In $g(x) = \sin(x + \frac{\pi}{3}) - 5$, the argument can be written as $(x - (-\frac{\pi}{3}))$, so $c = -\frac{\pi}{3}$, which corresponds to a shift to the left by $\frac{\pi}{3}$ units. The value of $d$ is $-5$, which corresponds to a vertical shift of 5 units down.
3. The graph of $g(x) = -5\cos(x)$ is obtained from the graph of $f(x) = \cos(x)$ by what sequence of transformations?
   Answer: A vertical stretch by a factor of 5, followed by a reflection over the x-axis. The function $g(x) = -5\cos(x)$ is of the form $y=af(x)$. The factor of 5 causes a vertical stretch by a factor of 5. The negative sign causes a reflection over the x-axis. The order of these two transformations does not matter.
4. How does the graph of $y = \sin(-3x)$ relate to the graph of $y = \sin(3x)$?
   Answer: The graph of $y = \sin(-3x)$ is a reflection of the graph of $y=\sin(3x)$ over the y-axis. A transformation of the form $y = f(-x)$ is a reflection of the graph of $y=f(x)$ over the y-axis. Therefore, the graph of $y = \sin(-3x)$ is a reflection of the graph of $y=\sin(3x)$ over the y-axis.
5. A sinusoidal function has an amplitude of 4, a period of $\pi$, a midline of $y=-1$, and a phase shift of $\frac{\pi}{2}$ to the right. Which of the following equations could represent this function?
   Answer: $y = 4\sin(2(x - \frac{\pi}{2})) - 1$ For a sinusoidal function $y = a\sin(b(x-c))+d$: The amplitude $|a|=4$. The midline $y=d$ is $y=-1$. The phase shift $c$ is $\frac{\pi}{2}$ to the right. The period is $\frac{2\pi}{|b|} = \pi$, which implies $|b|=2$. Combining these parameters gives the equation $y = 4\sin(2(x - \frac{\pi}{2})) - 1$.
6. The graph of a sinusoidal function is shifted down by 4 units. Which parameter in the general form $y = a\sin(b(x-c))+d$ is directly affected by this transformation?
   Answer: The parameter $d$, which represents the vertical shift. The parameter $d$ in the general form $y = a\sin(b(x-c))+d$ represents the vertical shift of the graph, which determines the midline $y=d$. A shift down by 4 units changes the value of $d$.
7. Which of the following statements correctly describes a characteristic of the function $f(x) = -3\sin(\frac{1}{2}x + \frac{\pi}{2}) - 1$?
   Answer: The amplitude is 3 and the midline is $y=-1$. For $f(x)=-3\sin(\frac{1}{2}x + \frac{\pi}{2}) - 1$: The amplitude is $|-3|=3$. The midline is $y=-1$. The period is $\frac{2\pi}{1/2}=4\pi$. The phase shift requires factoring: $\frac{1}{2}(x+\pi)$, so the shift is $\pi$ units to the left. The maximum value is $-1+3=2$ and the minimum is $-1-3=-4$. Therefore, the statement that the amplitude is 3 and the midline is $y=-1$ is correct.
8. The graph of $y = \cos(\frac{x}{3})$ is obtained from the graph of $y = \cos(x)$ by which of the following transformations?
   Answer: A horizontal stretch by a factor of 3 A transformation of the form $y = f(bx)$ results in a horizontal dilation of the graph of $y=f(x)$. If $|b| < 1$, it is a horizontal stretch. If $|b| > 1$, it is a horizontal compression. In $y = \cos(\frac{x}{3})$, $b=\frac{1}{3}$. Since $|\frac{1}{3}| < 1$, the transformation is a horizontal stretch by a factor of $1/b = 1/(1/3) = 3$.
9. The function $f(x) = \sin(x)$ can also be expressed as a phase-shifted cosine function. Which of the following expressions is equivalent to $f(x)$?
   Answer: $\cos(x - \frac{\pi}{2})$ The graph of $y=\cos(x)$ shifted to the right by $\frac{\pi}{2}$ units results in the graph of $y=\sin(x)$. This is a known trigonometric identity: $\sin(x) = \cos(x - \frac{\pi}{2})$.
10. Which of the following is the equation of the midline for the function $h(x) = 2\sin(\pi(x-1)) + 7$?
    Answer: $y = 7$ The midline of a sinusoidal function of the form $h(x) = a\sin(b(x-c))+d$ is the horizontal line $y = d$. For the function $h(x) = 2\sin(\pi(x-1)) + 7$, the value of $d$ is 7. Thus, the equation of the midline is $y = 7$.
11. What is the amplitude of the function $f(t) = 5 - 3\sin(4t + 1)$?
    Answer: $3$ The amplitude of a sinusoidal function of the form $f(t) = d + a\sin(b(t-c))$ is the absolute value of $a$. In the function $f(t) = 5 - 3\sin(4t + 1)$, the value of $a$ is $-3$. The amplitude is $|-3| = 3$.
12. The function $f(x)=\sin(x)$ is transformed to $g(x)=f(2x-\pi)$. Which of the following correctly describes the transformations?
    Answer: A horizontal compression by a factor of $\frac{1}{2}$ and a horizontal shift of $\frac{\pi}{2}$ units to the right. The transformed function is $g(x) = \sin(2x-\pi)$. To identify the transformations, we factor out the coefficient of $x$: $g(x) = \sin(2(x-\frac{\pi}{2}))$. This form shows a horizontal compression by a factor of $\frac{1}{2}$ (due to the 2 multiplying $x$) and a horizontal shift of $\frac{\pi}{2}$ units to the right (due to the $(x-\frac{\pi}{2})$).
13. What is the phase shift of the function $f(x) = -2\sin(4x+\pi)+1$?
    Answer: $\frac{\pi}{4}$ units to the left To find the phase shift, we must factor the coefficient of $x$ from the argument of the sine function. $4x+\pi = 4(x+\frac{\pi}{4})$. The function becomes $f(x) = -2\sin(4(x+\frac{\pi}{4}))+1$. This is of the form $y = a\sin(b(x-c))+d$ where $c = -\frac{\pi}{4}$. This represents a phase shift of $\frac{\pi}{4}$ units to the left.
14. The function $g(x)$ has twice the amplitude and half the period of the function $f(x) = 5\sin(2x) + 3$. The midline of $g(x)$ is the same as the midline of $f(x)$. Which of the following is an equation for $g(x)$?
    Answer: $g(x) = 10\sin(4x) + 3$ For $f(x) = 5\sin(2x) + 3$, the amplitude is 5, the period is $\frac{2\pi}{2} = \pi$, and the midline is $y=3$. For $g(x)$, the amplitude is twice that of $f(x)$, so $a=2 \cdot 5 = 10$. The period is half that of $f(x)$, so $P = \frac{\pi}{2}$. The period formula $P=\frac{2\pi}{b}$ gives $\frac{\pi}{2} = \frac{2\pi}{b}$, so $b=4$. The midline is the same, so $d=3$. Thus, $g(x) = 10\sin(4x) + 3$.
15. The graph of $y=\cos(x)$ is shifted $\frac{\pi}{6}$ units to the left, then reflected over the x-axis, and finally shifted 4 units up. Which of the following is the equation of the transformed function?
    Answer: $y = -\cos(x+\frac{\pi}{6}) + 4$ Let's apply the transformations in order. 1. Shift left by $\frac{\pi}{6}$ units: $y = \cos(x+\frac{\pi}{6})$. 2. Reflect over the x-axis (multiply the entire function by -1): $y = -\cos(x+\frac{\pi}{6})$. 3. Shift up by 4 units (add 4 to the function): $y = -\cos(x+\frac{\pi}{6}) + 4$.
16. A sinusoidal function has an amplitude of 5 and a midline of $y=2$. What are the maximum and minimum values of the function?
    Answer: Maximum value is 7, and minimum value is -3. The maximum value of a sinusoidal function is the midline plus the amplitude, and the minimum value is the midline minus the amplitude. Maximum Value = $d + |a| = 2 + 5 = 7$. Minimum Value = $d - |a| = 2 - 5 = -3$.
17. What is the period of the function $g(x) = -4\cos(\frac{\pi}{3}x + \pi) - 2$?
    Answer: $6$ The period of a sinusoidal function of the form $y = a\cos(b(x-c))+d$ is given by the formula $P = \frac{2\pi}{|b|}$. In the function $g(x) = -4\cos(\frac{\pi}{3}x + \pi) - 2$, the value of $b$ is $\frac{\pi}{3}$. Therefore, the period is $P = \frac{2\pi}{\pi/3} = 2\pi \cdot \frac{3}{\pi} = 6$.
18. What is the phase shift of the graph of the function $y = 3\cos(2x - \frac{\pi}{2})$?
    Answer: $\frac{\pi}{4}$ units to the right To determine the phase shift, the function must be written in the form $y = a\cos(b(x-c))+d$. We factor out the $b$ value from the argument: $2x - \frac{\pi}{2} = 2(x - \frac{\pi}{4})$. The equation becomes $y = 3\cos(2(x - \frac{\pi}{4}))$. Here, $c = \frac{\pi}{4}$, which represents a phase shift of $\frac{\pi}{4}$ units to the right.
19. A transformation is applied to the function $y=\cos(x)$ to change its period to $6\pi$ without altering its amplitude or midline. Which of the following functions represents this transformation?
    Answer: $y = \cos(\frac{1}{3}x)$ The period of $y = a\cos(bx)+d$ is $P = \frac{2\pi}{|b|}$. We are given that the new period is $6\pi$. So, $6\pi = \frac{2\pi}{|b|}$. Solving for $|b|$ gives $|b| = \frac{2\pi}{6\pi} = \frac{1}{3}$. The function is $y = \cos(\frac{1}{3}x)$.
20. A sinusoidal function $f(x) = a\sin(bx)+d$ has a maximum value of 10 and a minimum value of -2. What is the amplitude of the function?
    Answer: $6$ The amplitude of a sinusoidal function is half the difference between its maximum and minimum values. Amplitude = $\frac{\text{Maximum Value} - \text{Minimum Value}}{2}$. For this function, the amplitude is $\frac{10 - (-2)}{2} = \frac{12}{2} = 6$.

Start this quiz