Trig Waves Review Worksheet
Precalculus • Trigonometric Functions
Midline, Amplitude & Period Review
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1. Core Rules & Reference Formulas
Forms: \( y = A \sin(Bx) + D \) | \( y = A \cos(Bx) + D \)
Amplitude (\(|A|\)) Vertical Stretch
\( |A| = \frac{\text{Max} - \text{Min}}{2} \)
Half vertical distance peak-to-trough. Always positive: \( |A| > 0 \).
Midline (\(y = D\)) Vertical Shift
\( y = D = \frac{\text{Max} + \text{Min}}{2} \)
Horizontal center axis of oscillation. Always write as an equation: \( y = D \).
Period (\(T\)) Cycle Length
\( T = \frac{2\pi}{|B|} \quad \Leftrightarrow \quad |B| = \frac{2\pi}{T} \)
Horizontal span of 1 cycle. Key note: \( B \) is frequency parameter, not period.
2. Worked-Out Model: Step-by-Step Analysis
Guided Example
Model Problem: Analyze key features of the sinusoidal function:
\( f(x) = -3\cos\left(\frac{\pi}{2}x\right) + 1 \)
1. Identify Parameters
Compare with \( y = A\cos(Bx) + D \):
\( A = -3 \)
\( B = \frac{\pi}{2} \)
\( D = 1 \)
2. Amplitude
\( \text{Amp} = |A| = |-3| = \mathbf{3} \)
Distance is positive. The − indicates reflection over midline.
3. Midline & Period
Midline: \( \mathbf{y = 1} \)
Period:
\( T = \frac{2\pi}{B} = \frac{2\pi}{\pi/2} = 2\pi \cdot \frac{2}{\pi} = \mathbf{4} \)
4. Range & Extremes
\( \text{Max} = 1 + 3 = \mathbf{4} \)
\( \text{Min} = 1 - 3 = \mathbf{-2} \)
Range: \( [-2, 4] \)
3. Independent Practice: Apply the Rules
Show complete algebraic steps below
Question 1: Equation Analysis \( g(x) = 5\sin\left(\frac{2}{3}x\right) - 4 \)
Identify parameters and calculate all features. Show full work for the period fraction division.
a) Amplitude (\(|A|\)):
b) Midline Equation:
c) Period (\(T = \frac{2\pi}{B}\)):
d) Max & Min Values:
Question 2: Wave Modeling from Given Extremes Goal: \( y = A\cos(Bx) + D \)
A sinusoidal wave has a maximum of \( 9 \) at \( x = 0 \) and its first consecutive minimum of \( -1 \) at \( x = 6 \).
Part A: Midline & Amplitude Use formulas for \( D \) and \( |A| \):
Part B: Period (\(T\)) & Parameter \(B\) Half cycle = \( 6 \) units; find \( T \) & \( B = \frac{2\pi}{T} \):
Part C: Cosine Function Equation Assemble \( y = A\cos(Bx) + D \):
Reminder: Did you write midline as \( y = \dots \)? Amplitude is non-negative. \( B \) is the frequency factor, not the period. Precalculus • Periodic Functions Review
Trig Waves Answer Key
Teacher Resource • Solutions Guide Precalculus
Trig Waves Review • Answer Key & Steps
Full Solutions & Rubric
Fast Grading Key:
Q1: Amp = 5 • \( y = -4 \) • \( T = 3\pi \) • Max = 1, Min = -9
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Q2: \( D = 4 \) (\( y = 4 \)) • \( |A| = 5 \) • \( T = 12 \) • \( B = \frac{\pi}{6} \) • \( y = 5\cos\left(\frac{\pi}{6}x\right) + 4 \)
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Question 1: Equation Analysis of \( g(x) = 5\sin\left(\frac{2}{3}x\right) - 4 \)
5 Points Total
a) Amplitude 1 pt
\( \text{Amplitude} = \mathbf{5} \)
\( |A| = |5| = 5 \). Distance from midline to crest is 5 units. Always positive.
b) Midline 1 pt
\( \mathbf{y = -4} \)
\( D = -4 \). Must be written as the equation of a horizontal line, \( y = -4 \) (not just \(-4\)).
c) Period (\(T\)) 1.5 pts
\( T = \mathbf{3\pi} \)
\( T = \frac{2\pi}{B} = \frac{2\pi}{2/3} = 2\pi \cdot \frac{3}{2} = 3\pi \). Multiply by reciprocal.
d) Max & Min 1.5 pts
\( \text{Max} = \mathbf{1}, \; \text{Min} = \mathbf{-9} \)
\( \text{Max} = -4 + 5 = 1 \)
\( \text{Min} = -4 - 5 = -9 \)
Range is \( [-9, 1] \).
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Question 2: Wave Modeling from Maximum \((0, 9)\) and Minimum \((6, -1)\)
5 Points Total
Part A: Midline & Amplitude 1.5 pts
Midline (\(D\)):
\( D = \frac{9 + (-1)}{2} = \frac{8}{2} = 4 \implies \mathbf{y = 4} \)
Amplitude (\(|A|\)):
\( |A| = \frac{9 - (-1)}{2} = \frac{10}{2} = \mathbf{5} \)
Part B: Period & Frequency Factor 1.5 pts
Distance from peak (\(x = 0\)) to trough (\(x = 6\)) is a half cycle:
\( \frac{1}{2}T = 6 \implies T = 2(6) = \mathbf{12} \)
Solve for \( B \):
\( B = \frac{2\pi}{T} = \frac{2\pi}{12} = \mathbf{\frac{\pi}{6}} \)
Part C: Cosine Function Equation 2 pts
Since the function starts at its maximum at \( x = 0 \), standard cosine requires no phase shift (\(C = 0\)) and positive amplitude (\(A = 5\)).
\( \mathbf{y = 5\cos\left(\frac{\pi}{6}x\right) + 4} \)
Verification check: \( f(0) = 5(1) + 4 = 9 \)✓; \( f(6) = 5(-1) + 4 = -1 \)✓.
Teacher Diagnostic & Common Pitfalls
1. Half-Cycle Period Mistake
Students often set \( T = 6 \) (the distance between max and min). Remind students: peak-to-trough is only half a cycle (\(\frac{1}{2}T\)), so full cycle length is \( 2 \times 6 = 12 \).
2. B vs. Period Confusion
In Q1, students may claim the period is \( \frac{2}{3} \). Reinforce that \( B \) is the coefficient inside the argument, and \( T = \frac{2\pi}{B} = 2\pi \cdot \frac{3}{2} = 3\pi \).
3. Midline Notation Error
Writing just "-4" or "4" loses points. Emphasize that the midline is a geometric locus (a horizontal line) requiring an equation: \( y = -4 \) or \( y = 4 \).
Grading Rubric: 1 pt per correct characteristic • 1 pt for accurate work/steps • 2 pts for correct final equation. Total: 10 Points. Precalculus • Trigonometry Unit Review
