Precalculus Flashcards: Modeling Periodic Phenomena With Trigonometric Functions

What this deck covers

This deck focuses on Modeling Periodic Phenomena With Trigonometric Functions, giving you a quick way to review the definitions, rules, and examples that matter most for Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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