Modeling Periodic Phenomena with Trigonometric Functions - Precalculus

1. Choose the correct phase shift $C$ for $y=4\sin\left(\frac{\pi}{2}(x-3)\right)+1$.

   $C=3$. Phase shift is the value subtracted from $x$ in the argument.

2. Which function starts at a maximum when $A>0$ and $C=0$: sine or cosine?

   cosine. $\cos(0) = 1$, so cosine starts at its maximum when $A > 0$.

3. What is the vertical shift of $y=A\sin(Bx)+D$?

   $D$ units (midline $y=D$). The $+D$ term shifts the entire graph up by $D$ units.

4. What is the horizontal phase shift of $y=A\cos(B(x-C))+D$?

   $C$ units to the right. The $(x-C)$ form shifts the graph $C$ units right.

5. Identify a model for max $10$, min $2$, and period $12$ using cosine with no phase shift.

   $y=4\cos\left(\frac{\pi}{6}x\right)+6$. Amp $= \frac{10-2}{2} = 4$, midline $= \frac{10+2}{2} = 6$, $B = \frac{2\pi}{12} = \frac{\pi}{6}$.

6. Given max $M$ and min $m$ of a sinusoid, what is the midline value $D$?

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

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

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

8. State the standard sinusoid model (using sine) with amplitude $A$, period $P$, midline $D$, and phase shift $C$.

   $y=A\sin\left(\frac{2\pi}{P}(x-C)\right)+D$. Standard form where $B=\frac{2\pi}{P}$ relates period to angular frequency.

9. State the standard sinusoid model (using cosine) with amplitude $A$, period $P$, midline $D$, and phase shift $C$.

   $y=A\cos\left(\frac{2\pi}{P}(x-C)\right)+D$. Same as sine form but uses cosine; $B=\frac{2\pi}{P}$ converts period to angular frequency.

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

    $y=D$. The midline is the vertical shift, the constant $D$ added to the function.

11. Write a cosine model with amplitude $2$, period $6$, midline $4$, and a maximum at $x=1$.

    $y=2\cos\left(\frac{\pi}{3}(x-1)\right)+4$. Cosine has max at $x=C$ when no reflection; $B=\frac{2\pi}{6}=\frac{\pi}{3}$.

12. Identify the amplitude of $y=-2\cos\left(\frac{\pi}{4}(x+1)\right)+7$.

    $2$. Amplitude is $|-2|=2$, the absolute value of the coefficient.

13. What is the period of $y=A\sin\left(\frac{2\pi}{P}(x-C)\right)+D$ in terms of $P$?

    $P$. When $B=\frac{2\pi}{P}$, the period is simply $P$.

14. What is the standard sinusoid form (using sine) that shows amplitude, midline, and period?

    $y=A\sin(B(x-C))+D$. Standard form where $A$ is amplitude, $B$ affects period, $C$ is phase shift, $D$ is midline.

15. What is the standard sinusoid form (using cosine) that shows amplitude, midline, and period?

    $y=A\cos(B(x-C))+D$. Standard form where $A$ is amplitude, $B$ affects period, $C$ is phase shift, $D$ is midline.

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

    $|B|$. Frequency counts how many complete cycles occur in $2\pi$ units.

17. Find the amplitude, midline, and period of $y=-4\cos\left(\frac{\pi}{3}x\right)+7$.

    amp $=4$, midline $y=7$, period $=6$. $|A| = 4$, $D = 7$, period $= \frac{2\pi}{\frac{\pi}{3}} = 6$.

18. Identify the midline of $y=-2\cos\left(\frac{\pi}{4}(x+1)\right)+7$.

    $y=7$. The midline is the constant term added outside the trig function.

19. Identify the max and min values of $y=A\sin(Bx)+D$ using $A$ and $D$.

    max $=D+|A|$, min $=D-|A|$. Max occurs when sine equals 1, min when sine equals -1.

20. Write a sine model with amplitude $3$, period $10$, midline $-2$, and no phase shift.

    $y=3\sin\left(\frac{\pi}{5}x\right)-2$. $B=\frac{2\pi}{10}=\frac{\pi}{5}$ for period 10; no phase shift means no $(x-C)$ term.

21. Choose $B$ if the period is $8$ for $y=A\sin(Bx)+D$.

    $B=\frac{\pi}{4}$. Since $P=\frac{2\pi}{B}=8$, solving gives $B=\frac{\pi}{4}$.

22. Identify the period of $y=-2\cos\left(\frac{\pi}{4}(x+1)\right)+7$.

    $8$. $P=\frac{2\pi}{\pi/4}=8$ using the period formula.

23. What is the frequency (cycles per unit) of $y=A\sin\left(\frac{2\pi}{P}(x-C)\right)+D$?

    $\frac{1}{P}$. Frequency is reciprocal of period: $f=\frac{1}{P}$ cycles per unit.

24. Given max $M$ and min $m$ of a sinusoid, what is the amplitude?

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

25. Find a sine model with amplitude $3$, midline $y=2$, and period $4\pi$.

    $y=3\sin\left(\frac{1}{2}x\right)+2$. Use $B = \frac{2\pi}{4\pi} = \frac{1}{2}$ for period $4\pi$.

26. Find a cosine model with amplitude $5$, midline $y=-1$, and period $\pi$.

    $y=5\cos(2x)-1$. Use $B = \frac{2\pi}{\pi} = 2$ for period $\pi$.

27. Find the period and frequency of a sinusoid with $B=\frac{\pi}{3}$ in $y=A\sin(Bx)+D$.

    $P=6,\ f=\frac{1}{6}$. $P=\frac{2\pi}{\pi/3}=6$, and frequency $f=\frac{1}{6}$.

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

    $y=D$. The midline is the horizontal line around which the function oscillates.

29. Which function starts at the midline and increases when $A>0$ and $C=0$: sine or cosine?

    sine. $\sin(0) = 0$ starts at midline; $\cos(0) = 1$ starts at maximum.

30. What is the amplitude of a sinusoid with maximum $M$ and minimum $m$?

    $\frac{M-m}{2}$. Amplitude is half the distance between maximum and minimum values.

31. Choose $A$ and $D$ so the sinusoid has max $9$ and min $-3$ (no other changes).

    $A=6$, $D=3$. Amp $= \frac{9-(-3)}{2} = 6$, midline $= \frac{9+(-3)}{2} = 3$.

32. What is the midline value $D$ of a sinusoid with maximum $M$ and minimum $m$?

    $\frac{M+m}{2}$. Midline is the average of the maximum and minimum values.

33. What is $B$ if a sinusoid has period $T$ (in radians) in $y=A\sin(Bx)+D$?

    $B=\frac{2\pi}{T}$. To get period $T$, set $B = \frac{2\pi}{T}$ in the standard form.

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

    $\frac{2\pi}{|B|}$. Period formula divides $2\pi$ by the absolute value of the coefficient of $x$.

35. Identify the angular frequency $\omega$ in $y=A\sin(\omega(x-C))+D$.

    $\omega=|B|$. Angular frequency is the coefficient of $(x-C)$, which is $|B|$.

36. Find the amplitude and midline of a sinusoid with maximum $10$ and minimum $2$.

    $A=4,\ D=6$. $A=\frac{10-2}{2}=4$, $D=\frac{10+2}{2}=6$ using the formulas.

37. Write a cosine model with amplitude $5$, period $12$, midline $1$, and no phase shift.

    $y=5\cos\left(\frac{\pi}{6}x\right)+1$. $B=\frac{2\pi}{12}=\frac{\pi}{6}$ for period 12; cosine with no phase shift.

38. What is the period $T$ if a sinusoid completes $f$ cycles per unit on the $x$-axis?

    $T=\frac{1}{f}$. Period is the reciprocal of frequency.

Study these flashcards