Sinusoidal Models (IB Precalculus)- Homework 10 Solutions
Sinusoidal Models in IB Precalculus: Homework 10 Deep Dive
If you're grinding through Homework 10 and hitting a wall with sinusoidal models, you're not alone. These problems trip up even the strongest math students. This guide cuts through the nonsense and gives you what you actually need to solve these problems.
What Are Sinusoidal Models?
Sinusoidal models describe anything that moves up and down in a regular pattern. Waves. Pendulums. Seasonal temperatures. AC electricity. The sine and cosine functions are your tools here.
The general equation you'll see everywhere:
f(t) = A sin(B(t - C)) + D
or
f(t) = A cos(B(t - C)) + D
Each letter controls something specific. Memorize this or you're dead in the water.
Breaking Down the Parameters
- A = Amplitude. How far the graph stretches vertically from its midline. Half the total distance from peak to trough.
- B = Controls the period. Period = 2Ο/B. Bigger B means faster oscillation, shorter period.
- C = Horizontal phase shift. Moves the graph left or right. Called the phase displacement in IB language.
- D = Vertical shift. Moves everything up or down. The midline sits at y = D.
Common Homework 10 Problem Types
1. Finding the Equation from a Graph
You'll get a sine or cosine wave and need to extract the four parameters. Here's your checklist:
- Count the vertical distance from max to min, divide by 2 β amplitude
- Find the period by measuring one full wave cycle β B = 2Ο/period
- Locate where the graph crosses its midline going upward β that x-value gives you C (for sine)
- The midline height is your D
2. Modeling Real-World Situations
Temperature cycles, tides, ferris wheelsβ IB loves these. The trick: identify what the period should represent in real terms.
Example: "Temperature varies from 50Β°F to 90Β°F over a 24-hour cycle"
- Amplitude = (90 - 50)/2 = 20
- Midline = (90 + 50)/2 = 70
- Period = 24 hours, so B = 2Ο/24 = Ο/12
- Peak occurs around hour 15 (afternoon), so you need to shift accordingly
3. Writing Models with Specific Conditions
Sometimes they give you exact points the function must pass through. You solve for A, B, C, and D using systems of equations. Brutal if you're not comfortable with algebra, straightforward if you are.
Step-by-Step: Solving a Typical Problem
Problem: A Ferris wheel with radius 20 meters makes one complete rotation every 60 seconds. The lowest seat sits 2 meters above ground. Write a sinusoidal model for the height of a seat above ground, starting when the seat is at the lowest point.
Step 1: Identify your baseline values
Radius = 20 meters. Center of wheel = 20 + 2 = 22 meters above ground. This is your midline, so D = 22.
Step 2: Find amplitude
A = 20 (radius equals amplitude for this setup)
Step 3: Find period and B
One rotation = 60 seconds. Period = 60. B = 2Ο/60 = Ο/30
Step 4: Determine phase shift and function type
We start at the lowest point. Cosine starts at maximum, sine starts at midline going up. Neither fits perfectly, but we can use a negative cosine or shift sine appropriately.
Lowest point = minimum. We want the function to equal D - A at t = 0.
h(t) = -20 cos(Οt/30) + 22
Check: at t = 0, cos(0) = 1, so h = -20(1) + 22 = 2 β
Key Formulas Reference Table
| Parameter | What It Controls | How to Find It |
|---|---|---|
| A (Amplitude) | Vertical stretch | (max - min) / 2 |
| B (Angular frequency) | Period length | 2Ο / period |
| C (Phase shift) | Horizontal displacement | Offset from standard position |
| D (Vertical shift) | Midline height | (max + min) / 2 |
Phase Shift: The Part Students Screw Up Most
The horizontal shift C is where most people lose marks. The formula shows (t - C), not (t + C).
If C is positive, the graph shifts right.
If C is negative, the graph shifts left.
Watch this: f(t) = sin(t - Ο/4) shifts right by Ο/4. f(t) = sin(t + Ο/4) shifts left by Ο/4.
IB textbooks sometimes write it as sin(Bt - BC), which looks different but means the same thing. Factor out the B to see the actual shift.
Sin vs Cos: Which One Do I Use?
Honestly? Either works if you adjust the phase shift correctly. But one is usually cleaner.
- Use cosine when the graph starts at a maximum or minimum
- Use sine when the graph starts at the midline crossing upward
If you pick wrong, you'll just get a messier equation. It's not wrong, just inefficient. IB graders notice this.
Common Mistakes That Cost You Points
- Forgetting to divide by 2 when calculating amplitude from max/min values
- Getting B backwards β remember period = 2Ο/B, not B/2Ο
- Ignoring units β if time is in minutes but the problem implies hours, fix it
- Not checking your answer β plug in known points and verify
- Using sine when cosine is cleaner β doesn't earn you bonus points for extra work
Quick Checklist Before You Submit
- β Does your amplitude match half the vertical range?
- β Does your period produce the correct frequency?
- β Does the function hit the right values at t = 0 (or whatever starting point)?
- β Did you use the correct trig function or compensate with phase shift?
- β Are your units consistent throughout?
When You're Stuck
Draw it. Sketch the graph. Label the max, min, midline, and period. Visual learners solve these faster because the math becomes obvious once you see the shape.
If you can't sketch it, at least identify: where does the cycle start? Where does it end? Where's the middle?
These three points tell you almost everything you need for the parameters.
That's it. No inspirational ending. Go do your homework.