How to Solve: Trigonometric Graphs

How to Solve: Trigonometric Graphs

Introduction

"Master trigonometric graphs, and you’ll predict ocean tides, design roller coasters, and crush every exam question on transformations—guaranteed."

What You Need To Know First

1. Basic trigonometric functions: You must know the shapes of y = sin(x), y = cos(x), and y = tan(x) graphs, including their key points (e.g., sin(0) = 0, cos(0) = 1).
2. Transformations: Understand how a, b, c, and d affect y = a·sin(bx + c) + d (vertical stretch, horizontal shift, etc.).
3. Period and amplitude: Know that amplitude = |a|, period = 2π/|b| for sine/cosine, and period = π/|b| for tangent.

If you’re shaky on any of these, pause and review them first.

Key Vocabulary

Formulas To Know

1. General Form of Trigonometric Functions

Formula: y = a·sin(bx + c) + d y = a·cos(bx + c) + d y = a·tan(bx + c) + d

Variables: - a = amplitude (vertical stretch/compression; if negative, reflects over midline). - b = affects period (horizontal stretch/compression). - c = phase shift (horizontal shift). - d = vertical shift (moves midline up/down).

Memorise? ✅ MEMORISE THIS (not always given on exams).

2. Period Formulas

For sine and cosine: Period = 2π / |b|

For tangent: Period = π / |b|

Memorise? ✅ MEMORISE THIS.

3. Phase Shift Formula

Phase Shift = -c / b - If c/b is positive, shift left. - If c/b is negative, shift right.

Memorise? ✅ MEMORISE THIS.

4. Key Points for One Period

For y = sin(x) and y = cos(x), one period includes these x-values (adjust for b and c): - 0, π/2, π, 3π/2, 2π

For y = tan(x), one period includes: - -π/2, -π/4, 0, π/4, π/2 (with asymptotes at -π/2 and π/2).

Memorise? ❌ Given on most exam sheets (but know how to use them).

Step-by-Step Method

How to Sketch y = a·sin(bx + c) + d (or cosine/tangent) in 6 Steps

1. Find the amplitude (a)
2. Amplitude = |a|.

3. If a is negative, the graph is reflected over the midline.

4. Find the period (2π/|b| for sin/cos, π/|b| for tan)

5. Divide the standard period (2π or π) by |b|.

6. Find the phase shift (-c/b)

7. If c/b is positive, shift left by that amount.

8. If c/b is negative, shift right by that amount.

9. Find the vertical shift (d)

10. The midline moves to y = d.

11. Max = d + |a|, Min = d - |a|.

12. Plot key points for one period

13. For sin/cos: Start at x = -c/b (phase shift), then add period/4 for each key point.

14. For tan: Start at x = -c/b - period/2, then add period/4 for each point (skip asymptotes).

15. Draw the graph

16. For sin/cos: Smooth wave through key points.
17. For tan: Draw asymptotes at x = -c/b ± period/2, then sketch the curve between them.

Worked Example Using the Steps

Problem: Sketch y = 2sin(πx - π/2) + 1 for one period.

Step 1: Amplitude - a = 2 → Amplitude = 2.

Step 2: Period - b = π → Period = 2π / π = 2.

Step 3: Phase Shift - c = -π/2 → Phase Shift = -(-π/2)/π = 1/2 → Shift right by 1/2.

Step 4: Vertical Shift - d = 1 → Midline = y = 1. - Max = 1 + 2 = 3, Min = 1 - 2 = -1.

Step 5: Key Points - Start at x = 1/2 (phase shift). - Add period/4 = 2/4 = 0.5 for each point: - x = 1/2 → y = 2sin(π(1/2) - π/2) + 1 = 2sin(0) + 1 = 1 - x = 1 → y = 2sin(π - π/2) + 1 = 2sin(π/2) + 1 = 3 - x = 1.5 → y = 2sin(3π/2 - π/2) + 1 = 2sin(π) + 1 = 1 - x = 2 → y = 2sin(2π - π/2) + 1 = 2sin(3π/2) + 1 = -1 - x = 2.5 → y = 2sin(5π/2 - π/2) + 1 = 2sin(2π) + 1 = 1

Step 6: Draw the Graph - Plot points: (0.5, 1), (1, 3), (1.5, 1), (2, -1), (2.5, 1). - Draw a smooth sine wave through them.

What we did and why: We used the general form to extract a, b, c, and d, then calculated amplitude, period, and shifts. Plotting key points ensures accuracy, and the midline helps set the vertical range.

Worked Examples

Example 1 - Basic

Problem: Sketch y = cos(2x) for one period.

Step 1: Amplitude - a = 1 → Amplitude = 1.

Step 2: Period - b = 2 → Period = 2π/2 = π.

Step 3: Phase Shift - c = 0 → Phase Shift = 0.

Step 4: Vertical Shift - d = 0 → Midline = y = 0.

Step 5: Key Points - Start at x = 0, add π/4 for each point: - x = 0 → y = cos(0) = 1 - x = π/4 → y = cos(π/2) = 0 - x = π/2 → y = cos(π) = -1 - x = 3π/4 → y = cos(3π/2) = 0 - x = π → y = cos(2π) = 1

Step 6: Draw the Graph - Plot points and connect with a smooth cosine wave.

What we did and why: We identified the period change (b = 2 halves the period) and plotted key points at π/4 intervals to capture the compressed wave.

Example 2 - Medium

Problem: Sketch y = -3cos(x + π/4) - 2 for one period.

Step 1: Amplitude - a = -3 → Amplitude = 3 (negative reflects the graph).

Step 2: Period - b = 1 → Period = 2π/1 = 2π.

Step 3: Phase Shift - c = π/4 → Phase Shift = -π/4 / 1 = -π/4 → Shift left by π/4.

Step 4: Vertical Shift - d = -2 → Midline = y = -2. - Max = -2 + 3 = 1, Min = -2 - 3 = -5.

Step 5: Key Points - Start at x = -π/4, add π/2 for each point: - x = -π/4 → y = -3cos(0) - 2 = -5 - x = π/4 → y = -3cos(π/2) - 2 = -2 - x = 3π/4 → y = -3cos(π) - 2 = 1 - x = 5π/4 → y = -3cos(3π/2) - 2 = -2 - x = 7π/4 → y = -3cos(2π) - 2 = -5

Step 6: Draw the Graph - Plot points and connect with a reflected cosine wave.

What we did and why: The negative a flips the graph upside down, and the vertical shift moves the midline down. The phase shift moves the graph left, so we adjusted the starting x-value accordingly.

Example 3 - Exam Style

Problem: The height h (in meters) of a Ferris wheel seat above the ground is given by h = 10sin(πt/30 + π/6) + 12, where t is time in seconds. a) What is the maximum height of the seat? b) How long does one full rotation take? c) Sketch the graph for 0 ≤ t ≤ 60.

Part a) Maximum Height - Amplitude = 10, Midline = 12 → Max = 12 + 10 = 22 meters.

Part b) Period - b = π/30 → Period = 2π / (π/30) = 60 seconds.

Part c) Sketch the Graph Step 1: Amplitude - a = 10 → Amplitude = 10.

Step 2: Period - Period = 60 (from part b).

Step 3: Phase Shift - c = π/6 → Phase Shift = -π/6 / (π/30) = -5 → Shift left by 5 seconds.

Step 4: Vertical Shift - d = 12 → Midline = y = 12.

Step 5: Key Points - Start at t = -5 (but we only need 0 ≤ t ≤ 60), so adjust: - At t = 0: h = 10sin(π/6) + 12 = 10(0.5) + 12 = 17 - At t = 15: h = 10sin(π/2 + π/6) + 12 = 10sin(2π/3) + 12 ≈ 10(0.866) + 12 ≈ 20.66 - At t = 30: h = 10sin(π + π/6) + 12 = 10(-0.5) + 12 = 7 - At t = 45: h = 10sin(3π/2 + π/6) + 12 = 10(-1) + 12 = 2 - At t = 60: h = 10sin(2π + π/6) + 12 = 10(0.5) + 12 = 17

Step 6: Draw the Graph - Plot points: (0, 17), (15, 20.66), (30, 7), (45, 2), (60, 17). - Connect with a smooth sine wave.

What we did and why: We treated the real-world problem like a standard trig graph, using the given equation to extract amplitude, period, and shifts. The phase shift told us the graph starts 5 seconds "early," so we adjusted our key points accordingly.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, listen up—this is your 60-second cheat sheet for trig graphs. First, memorise the general form: y = a·sin(bx + c) + d. Amplitude is |a|, period is 2π/|b| for sin/cos, π/|b| for tan. Phase shift is -c/b—if it’s positive, shift left; negative, shift right. Vertical shift is d, and that’s your midline. Plot key points at period/4 intervals, starting from the phase shift. For tan, remember asymptotes at x = -c/b ± period/2. Common traps? Mixing up phase shift signs, forgetting vertical shifts, and miscalculating periods. Double-check your work: does the graph start where it should? Does it repeat at the right interval? Nail this, and you’ll own every trig graph question on the exam. Now go practice!