Here are the solutions to the problems on your worksheet.
Part 1: Determine Amplitude and Period
Rule: For a function like $y = A \sin(Bx)$ or $y = A \cos(Bx)$:
*
Amplitude is the absolute value of $A$ ($|A|$). It tells you how tall the wave is from the center line.
*
Period is $\frac{2\pi}{|B|}$. It tells you how long one full cycle of the wave is.
1. $y = \sin 4x$
* $A = 1$, so Amplitude =
1.
* $B = 4$, so Period = $\frac{2\pi}{4}$ =
$\frac{\pi}{2}$.
2. $y = \cos 5x$
* $A = 1$, so Amplitude =
1.
* $B = 5$, so Period =
$\frac{2\pi}{5}$.
3. $y = 2 \sin x$
* $A = 2$, so Amplitude =
2.
* $B = 1$, so Period = $\frac{2\pi}{1}$ =
$2\pi$.
4. $y = -4 \sin 3x$
* $A = -4$. Amplitude is always positive, so $|-4|$ =
4.
* $B = 3$, so Period =
$\frac{2\pi}{3}$.
5. $y = 2 \sin (-4x)$
* $A = 2$, so Amplitude =
2.
* $B = -4$. We use the absolute value for period, so $|-4| = 4$. Period = $\frac{2\pi}{4}$ =
$\frac{\pi}{2}$.
6. $y = 3 \sin \frac{2}{3} x$
* $A = 3$, so Amplitude =
3.
* $B = \frac{2}{3}$. Period = $\frac{2\pi}{2/3}$. To divide by a fraction, multiply by the reciprocal: $2\pi \cdot \frac{3}{2} = 3\pi$. Period =
$3\pi$.
---
Part 2: Graph Analysis
7.
*
Amplitude: The graph goes up to 3 and down to -3. Amplitude =
3.
*
Period: One full wave starts at 0 and ends at $\pi$. Period =
$\pi$.
*
Equation: Since it starts at (0,0) and goes up, it is sine.
* $a = 3$.
* $b = \frac{2\pi}{\text{period}} = \frac{2\pi}{\pi} = 2$.
* Equation:
$y = 3 \sin 2x$
8.
*
Amplitude: The graph goes up to 4 and down to -4. Amplitude =
4.
*
Period: One full wave starts at 0 and ends at $2\pi$. Period =
$2\pi$.
*
Equation: It starts at the maximum height (0, 4), so it is cosine.
* $a = 4$.
* $b = \frac{2\pi}{2\pi} = 1$.
* Equation:
$y = 4 \cos x$
9.
*
Amplitude: The graph goes up to 2 and down to -2. Amplitude =
2.
*
Period: One full wave starts at 0 and ends at $4\pi$. Period =
$4\pi$.
*
Equation: Starts at (0,0) and goes up, so it is sine.
* $a = 2$.
* $b = \frac{2\pi}{4\pi} = \frac{1}{2}$.
* Equation:
$y = 2 \sin \frac{1}{2}x$
10.
*
Amplitude: The graph goes up to 5 and down to -5. Amplitude =
5.
*
Period: One full wave starts at 0 and ends at $2\pi$. Period =
$2\pi$.
*
Equation: It starts at the minimum height (0, -5), so it is a negative cosine.
* $a = -5$.
* $b = \frac{2\pi}{2\pi} = 1$.
* Equation:
$y = -5 \cos x$
---
Part 3: Sketching Graphs
*(Note: I cannot draw on your paper, but here is exactly where to plot the points)*
11. $y = 4 \sin x$
*
Key Points to Plot:
* Start at
(0, 0).
* Go up to max at
($\frac{\pi}{2}, 4$).
* Cross middle at
($\pi, 0$).
* Go down to min at
($\frac{3\pi}{2}, -4$).
* End cycle at
($2\pi, 0$).
* Repeat backwards for negative x:
($-\frac{\pi}{2}, -4$),
($-\pi, 0$), etc.
12. $y = 2 \cos x$
*
Key Points to Plot:
* Start at max at
(0, 2).
* Cross middle at
($\frac{\pi}{2}, 0$).
* Go down to min at
($\pi, -2$).
* Cross middle at
($\frac{3\pi}{2}, 0$).
* End cycle at
($2\pi, 2$).
* Cosine is symmetric, so the left side mirrors the right side starting from (0,2).
Final Answer:
1. Amp: 1, Per: $\frac{\pi}{2}$
2. Amp: 1, Per: $\frac{2\pi}{5}$
3. Amp: 2, Per: $2\pi$
4. Amp: 4, Per: $\frac{2\pi}{3}$
5. Amp: 2, Per: $\frac{\pi}{2}$
6. Amp: 3, Per: $3\pi$
7. Amp: 3, Per: $\pi$, Eq: $y = 3 \sin 2x$
8. Amp: 4, Per: $2\pi$, Eq: $y = 4 \cos x$
9. Amp: 2, Per: $4\pi$, Eq: $y = 2 \sin \frac{1}{2}x$
10. Amp: 5, Per: $2\pi$, Eq: $y = -5 \cos x$
11 & 12. See plotting instructions above.
Parent Tip: Review the logic above to help your child master the concept of graphs of sine and cosine functions worksheet.
