Modeling Periodic Phenomena with Trigonometric Functions - Geometry

1. Identify $A$, $B$, and $D$ for $y = -3\cos(\frac{\pi}{2}x)+4$.

   $A=-3,\ B=\frac{\pi}{2},\ D=4$. Direct identification of parameters from standard form.

2. What is the value of $B$ if $y = A\sin(Bx)+D$ has period $\pi$?

   $B = 2$. From period $\pi = \frac{2\pi}{B}$, so $B = 2$.

3. What is the amplitude of a periodic model with max $12$ and min $-4$?

   $8$. Amplitude = $\frac{12-(-4)}{2} = 8$.

4. What is the amplitude of $y = 5\sin(\frac{2\pi}{7}x)+1$?

   $5$. Amplitude is coefficient $5$.

5. What is the period of $y = 5\sin(\frac{2\pi}{7}x)+1$?

   $7$. Period = $\frac{2\pi}{\frac{2\pi}{7}} = 7$.

6. Choose $\sin$ or $\cos$: a model that is at its maximum at $x=0$ with $A>0$ and no shift.

   $\cos$. Cosine starts at maximum when $A > 0$.

7. What is the period of $y = A\sin(\frac{\pi}{6}x)+D$?

   $12$. Period = $\frac{2\pi}{|\frac{\pi}{6}|} = \frac{2\pi}{\frac{\pi}{6}} = 12$.

8. A Ferris wheel height ranges from $3$ m to $23$ m. What amplitude should the model use?

   $10$. Amplitude = $\frac{23-3}{2} = 10$.

9. Identify the error: period of $y=\cos(5x)$ was stated as $5$. What is correct period?

   Period is $\frac{2\pi}{5}$. Period is $\frac{2\pi}{B}$, not $B$ itself.

10. What is the period of $y = 3\cos(4x)-2$?

    $\frac{\pi}{2}$. Period = $\frac{2\pi}{4} = \frac{\pi}{2}$.

11. What is the period of $y = A\sin(B(x-C))+D$ in terms of $B$?

    $\frac{2\pi}{|B|}$. Period formula: $\frac{2\pi}{|B|}$ for sine functions.

12. A sinusoid has period $\frac{2\pi}{3}$. What is $B$ in $y = A\cos(Bx)+D$?

    $B = 3$. From period $\frac{2\pi}{3} = \frac{2\pi}{B}$, so $B = 3$.

13. What is the angular frequency parameter in $y = A\sin(Bx)$ that controls period?

    $B$. Parameter $B$ controls how fast the function oscillates.

14. Choose a model form: starts at maximum at $t=0$, amplitude $6$, midline $10$, period $8$.

    $y = 6\cos(\frac{\pi}{4}t)+10$. Cosine starts at maximum, period $8$ gives $B = \frac{\pi}{4}$.

15. What is the value of $B$ if $y = A\cos(Bx)+D$ has period $8\pi$?

    $B = \frac{1}{4}$. From period $8\pi = \frac{2\pi}{B}$, so $B = \frac{1}{4}$.

16. What is the amplitude of $y = A \sin(B(x - C)) + D$ in terms of parameters?

    $|A|$. Amplitude is the absolute value of coefficient $A$.

17. What is the midline of $y = A\sin(B(x-C))+D$ written as an equation?

    $y = D$. The midline is the horizontal line $y = D$.

18. What is the midline equation of $y = 2\cos(3x)-4$?

    $y = -4$. Midline is $y = D = -4$.

19. Choose a model form: starts at midline increasing at $t=0$, amplitude $6$, midline $10$, period $8$.

    $y = 6\sin(\frac{\pi}{4}t)+10$. Sine starts at midline increasing, period $8$ gives $B = \frac{\pi}{4}$.

20. A Ferris wheel height ranges from $3$ m to $23$ m. What midline value should the model use?

    $13$. Midline = $\frac{23+3}{2} = 13$.

21. A tide varies from $2$ m to $8$ m with period $12$ h. What midline value should the model use?

    $5$. Midline = $\frac{8+2}{2} = 5$.

22. What is the midline equation for a periodic model with max $12$ and min $-4$?

    $y = 4$. Midline = $\frac{12+(-4)}{2} = 4$.

23. What is the period of $y = \sin(\frac{\pi}{3}x)$?

    $6$. Period = $\frac{2\pi}{\frac{\pi}{3}} = 6$.

24. What is the frequency (cycles per $2\pi$) of $y = A\sin(Bx)$ in terms of $B$?

    $|B|$. Frequency is $|B|$ cycles per $2\pi$ units.

25. What is the midline value $D$ if the maximum is $M$ and minimum is $m$?

    $\frac{M+m}{2}$. Midline value is the average of max and min.

26. What is the amplitude of $y = -7\sin(2x)+5$?

    $7$. Amplitude is $|-7| = 7$.

27. A wheel makes one revolution every $40$ s. What $B$ fits $y=A\cos(Bt)+D$?

    $B = \frac{\pi}{20}$. From period $40 = \frac{2\pi}{B}$, so $B = \frac{\pi}{20}$.

28. What is the period of $y = -3\cos(\frac{\pi}{2}x)+4$?

    $4$. Period = $\frac{2\pi}{\frac{\pi}{2}} = 4$.

29. What is the vertical shift $D$ in $y = A\sin(Bx)+D$ if the midline is $y=k$?

    $D = k$. Vertical shift $D$ equals the midline value $k$.

30. What amplitude $A$ is needed if a sinusoid has maximum $9$ and minimum $1$?

    $A = 4$. Amplitude = $\frac{9-1}{2} = 4$.

31. Identify the amplitude of $y = \frac{1}{2}\sin(x)-3$.

    $\frac{1}{2}$. Amplitude is the coefficient $\frac{1}{2}$.

32. What is the phase shift of $y = A\sin(B(x-C))+D$ in terms of $C$?

    $C$ units right. Phase shift moves graph $C$ units horizontally.

33. What is the range of $y = A\cos(B(x-C))+D$ written with $A$ and $D$?

    $[D-|A|,\ D+|A|]$. Range extends $|A|$ units above and below midline $D$.

34. What is the amplitude of $y = a\sin(x)+d$ if the max is $M$ and min is $m$?

    $\frac{M-m}{2}$. Amplitude is half the difference between max and min.

35. What midline value $D$ is needed if a sinusoid has maximum $9$ and minimum $1$?

    $D = 5$. Midline value = $\frac{9+1}{2} = 5$.

36. A model has amplitude $3$ and midline $y=2$. What are its max and min values?

    max $5$, min $-1$. Max = $2+3 = 5$, min = $2-3 = -1$.

37. A tide has period $12$ h. What $B$ fits $y=A\sin(Bt)+D$ with $t$ in hours?

    $B = \frac{\pi}{6}$. From period $12 = \frac{2\pi}{B}$, so $B = \frac{\pi}{6}$.

38. What is the range of $y = A\sin(B(x-C))+D$ written with $A$ and $D$?

    $[D-|A|,\ D+|A|]$. Range extends $|A|$ units above and below midline $D$.

39. A tide varies from $2$ m to $8$ m with period $12$ h. What amplitude should the model use?

    $3$. Amplitude = $\frac{8-2}{2} = 3$.

40. What is the period of $y = A\cos(B(x-C))+D$ in terms of $B$?

    $\frac{2\pi}{|B|}$. Period formula: $\frac{2\pi}{|B|}$ for cosine functions.

41. Which function starts at a maximum when $A>0$ and $C=0$: $\sin$ or $\cos$?

    $\cos$. Cosine starts at maximum when no phase shift.

42. A sinusoid has period $10$. What is $B$ in $y = A\sin(Bx)+D$?

    $B = \frac{\pi}{5}$. From period $10 = \frac{2\pi}{B}$, so $B = \frac{\pi}{5}$.

43. What is the midline equation of $y = 5\sin(\frac{2\pi}{7}x)+1$?

    $y = 1$. Midline is $y = D = 1$.

44. Identify the range of $y = -2\cos(x)-6$ using interval notation.

    $[-8,\ -4]$. Range: $[-6-2, -6+2] = [-8, -4]$.

45. What is the amplitude of $y = A\sin(B(x-C))+D$ in terms of parameters?

    $|A|$. Amplitude is the absolute value of coefficient $A$.

46. Identify the error: amplitude of $y = 2\sin(3x)+5$ was stated as $5$. What is correct?

    Amplitude is $2$. Amplitude is coefficient of trig function, not vertical shift.

47. Which function starts on the midline going upward when $A>0$ and $C=0$: $\sin$ or $\cos$?

    $\sin$. Sine starts at midline going upward when no phase shift.

48. Choose $\sin$ or $\cos$: a model that crosses the midline at $x=0$ increasing with $A>0$.

    $\sin$. Sine crosses midline increasing when $A > 0$.

49. What is the period of $y = \sin(2\pi x)$?

    $1$. Period = $\frac{2\pi}{2\pi} = 1$.

50. Identify the range of $y = 4\sin(x)+1$ using interval notation.

    $[-3,\ 5]$. Range: $[1-4, 1+4] = [-3, 5]$.

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