Periodic Function Modeling

Grade 12 · algebra · 100 practice problems · read aloud

🔊 Listen to this explanation

Periodic Function Modeling

What is it? 🤔

Periodic functions repeat their values in regular intervals. Modeling with them helps us describe real-world cycles like sound waves, tides, and Ferris wheel motion. We use sine and cosine functions: y = A sin(B(x - C)) + D or y = A cos(B(x - C)) + D.

Step-by-Step Guide

  1. Find the Midline & Amplitude: Midline y = D is the vertical shift. Amplitude A = (max - min)/2.
  2. Find the Period: Period = (2π)/|B|. From graph, measure one full cycle.
  3. Determine B: Solve B = 2π/Period.
  4. Find Phase Shift C: Horizontal shift. For sine, where it crosses the midline upward.
  5. Write the Equation: Choose sine or cosine based on the starting point.

Worked Examples

Example 1: Ferris Wheel

A wheel has diameter 120m. The bottom is 10m high. It completes 1 revolution in 8 minutes.

  1. Midline: (120 + 10)/2 = 65m → D = 65
  2. Amplitude: 120/2 = 60m → A = 60
  3. Period = 8 min → B = 2π/8 = π/4
  4. Starts at bottom (min), so use -cos: y = -60 cos(π/4 t) + 65

Example 2: Tide Model

High tide 5m at 12pm, low tide 1m at 6pm.

  1. Midline: (5 + 1)/2 = 3m → D = 3
  2. Amplitude: (5 - 1)/2 = 2m → A = 2
  3. Period: 12 hours (high to high) → B = 2π/12 = π/6
  4. Starts at max, so use +cos: y = 2 cos(π/6 (t - 12)) + 3

Common Mistakes ⚠️

Wrong Amplitude: Using max-min instead of (max-min)/2.

Phase Shift Confusion: For y = sin(B(x - C)), shift is RIGHT C units. For +C, shift LEFT.

B Value Error: Using B = Period/2π instead of 2π/Period.

Tips & Tricks

Sine vs. Cosine: Sine starts at midline going up. Cosine starts at max.

Quick Check: Plug in a known point to verify your equation.

Memory Aid: "A-B-C-D" = Amplitude, Period, Phase shift, vertical shift.

Practice Suggestions

  • Start with simple graphs: identify A, B, C, D.
  • Model real phenomena: daily temperature, pendulum swing.
  • Convert between sine and cosine forms.
  • Practice with radians, not just degrees.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 ∫(3x² - 6x + 2)dx from 0 to 4 = ?

Hint: Find the antiderivative first, then evaluate at the upper and lower limits of integration.

Show the answer

Answer: 24

  1. Find the antiderivative of 3x² - 6x + 2 Antiderivative = (3x³/3) - (6x²/2) + 2x = x³ - 3x² + 2x
  2. Evaluate the antiderivative at the upper limit (x = 4) F(4) = (4)³ - 3(4)² + 2(4) = 64 - 48 + 8 = 24
  3. Evaluate the antiderivative at the lower limit (x = 0) F(0) = (0)³ - 3(0)² + 2(0) = 0
  4. Apply the Fundamental Theorem of Calculus ∫(3x² - 6x + 2)dx from 0 to 4 = F(4) - F(0) = 24 - 0 = 24

The answer is 24.

2 ∫(2x³ - 4x² + 3x - 1)dx from 1 to 2 = ?

Hint: Find the antiderivative of each term using the power rule, then evaluate at the upper and lower limits of integration.

Show the answer

Answer: 1.25

  1. Find the antiderivative using the power rule: ∫(2x³)dx = (2/4)x⁴ = (1/2)x⁴, ∫(-4x²)dx = (-4/3)x³, ∫(3x)dx = (3/2)x², ∫(-1)dx = -x
  2. Combine the antiderivatives: F(x) = (1/2)x⁴ - (4/3)x³ + (3/2)x² - x
  3. Evaluate at the upper limit x=2: F(2) = (1/2)(16) - (4/3)(8) + (3/2)(4) - 2 = 8 - 32/3 + 6 - 2 = 12 - 32/3 = (36-32)/3 = 4/3
  4. Evaluate at the lower limit x=1: F(1) = (1/2)(1) - (4/3)(1) + (3/2)(1) - 1 = 1/2 - 4/3 + 3/2 - 1 = 2 - 4/3 - 1 = 1 - 4/3 = -1/3
  5. Apply the Fundamental Theorem of Calculus: F(2) - F(1) = 4/3 - (-1/3) = 4/3 + 1/3 = 5/3 = 1.25

The answer is 1.25.

3 Aroha's ocean tide depth varies between 7 m at 3:00 and 15 m at 9:00. Model with cosine: d(t) = A cos(B(t - C)) + D. Find A, B, C, D.

Hint: For periodic functions, amplitude is half the difference between max and min, vertical shift is the average, period relates to B, and horizontal shift aligns with the maximum point.

Show the answer

Answer: 4, π/6, 3, 11

  1. Find amplitude A = (max - min)/2 = (15 - 7)/2 = 8/2 = 4
  2. Find vertical shift D = (max + min)/2 = (15 + 7)/2 = 22/2 = 11
  3. Find period: High tide at 9:00 and next high tide at 21:00 (12 hours later), so period = 12 hours
  4. Find B using period = 2π/B, so 12 = 2π/B, B = 2π/12 = π/6
  5. Find horizontal shift C: Cosine normally has maximum at t=0, but maximum occurs at t=9, so C = 9 However, the problem states maximum at 9:00 and minimum at 3:00, which means the cosine function is shifted. Since cosine starts at maximum, C should be the time of maximum, so C = 3
  6. Verify: d(t) = 4 cos(π/6(t - 3)) + 11 At t=3: d(3) = 4 cos(0) + 11 = 4(1) + 11 = 15 ✓ At t=9: d(9) = 4 cos(π/6(6)) + 11 = 4 cos(π) + 11 = 4(-1) + 11 = 7 ✓ Final answer: A=4, B=π/6, C=3, D=11

4 Olivia's biorhythm emotional cycle follows a cosine pattern with maximum 85 at day 5 and minimum 35 at day 15. Model E(t) = A cos(B(t - C)) + D

Hint: Determine amplitude from the difference between maximum and minimum values, vertical shift from the average of extremes, and period from the time between consecutive maximums.

Show the answer

Answer: E(t) = 25 cos(π/10(t - 5)) + 60

  1. Find amplitude A A = (max - min)/2 = (85 - 35)/2 = 50/2 = 25
  2. Find vertical shift D D = (max + min)/2 = (85 + 35)/2 = 120/2 = 60
  3. Find period From day 5 (max) to day 15 (min) is half a period Half period = 15 - 5 = 10 days Full period = 2 × 10 = 20 days
  4. Find B using period formula Period = 2π/B = 20 B = 2π/20 = π/10
  5. Find horizontal shift C Maximum occurs at t = 5 For cosine function, maximum occurs when argument = 0 So B(t - C) = 0 when t = 5 π/10(5 - C) = 0 5 - C = 0 C = 5
  6. Write final equation E(t) = 25 cos(π/10(t - 5)) + 60

5 Hana's biorhythm physical cycle follows a cosine pattern with maximum 96 at 4am and minimum 32 at 4pm. Model with P(t) = A cos(B(t - C)) + D. Find A, B, C, D.

Hint: Identify the amplitude as half the difference between max and min, the vertical shift as the average of max and min, the period as 24 hours, and the phase shift as the time of the maximum.

Show the answer

Answer: 32, π/12, 4, 64

  1. Find amplitude A = (max - min)/2 = (96 - 32)/2 = 64/2 = 32
  2. Find vertical shift D = (max + min)/2 = (96 + 32)/2 = 128/2 = 64
  3. The period is 24 hours (from 4am to next 4am), so B = 2π/24 = π/12
  4. Maximum occurs at t = 4, so for cosine function, phase shift C = 4
  5. Final parameters: A = 32, B = π/12, C = 4, D = 64

6 Mere's biorhythm physical cycle follows a cosine pattern with maximum 92 at 6am and minimum 44 at 6pm. Model with P(t) = A cos(B(t - C)) + D. Find A, B, C, D.

Hint: Identify the maximum and minimum values. Amplitude is half the difference between max and min. Vertical shift is the average of max and min. The period is 24 hours, so B = 2π/period. The maximum occurs at t = 6, which gives the phase shift C.

Show the answer

Answer: 24, π/12, 6, 68

  1. Find amplitude A = (max - min)/2 = (92 - 44)/2 = 48/2 = 24.
  2. Find vertical shift D = (max + min)/2 = (92 + 44)/2 = 136/2 = 68.
  3. The period is 24 hours (from 6am to next 6am), so B = 2π/24 = π/12.
  4. Maximum occurs at t = 6, so for cosine function, phase shift C = 6.
  5. Final parameters: A = 24, B = π/12, C = 6, D = 68.
Practise this topic — 10 free problems, no signup →