Graphs of Trig Functions — ACT Math Guide
Graphs of trig functions ACT questions test your ability to read sine, cosine, and tangent curves and understand their transformations. These visual problems require you to identify amplitude, period, phase shifts, and vertical shifts from equations or graphs. The ACT math section typically includes 2-3 trigonometry questions out of 60 total questions, and with only 60 minutes to complete the test, knowing these patterns saves precious time. You'll master these graphs faster than you think once you learn the key patterns.
What You Need to Know
Sine and cosine graphs — both oscillate between -1 and 1 with period 2π
Amplitude — the height of the wave, found in y = A sin(x) or y = A cos(x)
Period — how long one complete cycle takes, calculated as 2π/B in y = sin(Bx)
Phase shift — horizontal movement left or right, found in y = sin(x - C)
Vertical shift — moves the entire graph up or down by constant D
Tangent graphs — have vertical asymptotes and period π (not 2π)
📐 KEY FORMULA: y = A sin(B(x - C)) + D or y = A cos(B(x - C)) + D
⏱️ ACT TIME TIP: Eliminate obviously wrong answers first — if amplitude is 3, answers showing waves reaching height 5 are wrong
How to Solve Graphs of Trig Functions on the ACT
Example Question 1 — Easy/Medium Difficulty
Which function has amplitude 2 and period π?
A) y = 2 sin(x)
B) y = sin(2x)
C) y = 2 sin(2x)
D) y = 2 cos(x) + 2
E) y = sin(x) + 2
Solution:
Step 1: Find amplitude — coefficient of sine or cosine function (A in y = A sin(x))
Step 2: Find period — use formula 2π/B where B is coefficient of x
Step 3: Check each option systematically
For amplitude 2: need coefficient 2 in front of sin or cos
For period π: need 2π/B = π, so B = 2
Option C: y = 2 sin(2x) has amplitude 2 and period 2π/2 = π
✅Answer: C — This function has both amplitude 2 and period π
Example Question 2 — Hard Difficulty
The graph of y = 3 cos(2x - π/2) + 1 has which of the following characteristics?
A) Amplitude 3, period π, phase shift π/4 right, vertical shift 1 up
B) Amplitude 3, period 2π, phase shift π/2 right, vertical shift 1 up
C) Amplitude 1, period π, phase shift π/4 right, vertical shift 3 up
D) Amplitude 3, period π, phase shift π/2 right, vertical shift 1 up
E) Amplitude 4, period π, phase shift π/4 right, vertical shift 1 up
Solution:
Step 1: Rewrite in standard form y = A cos(B(x - C)) + D
y = 3 cos(2(x - π/4)) + 1
Step 2: Identify each transformation
Amplitude |A| = 3
Period = 2π/B = 2π/2 = π
Phase shift C = π/4 right
Vertical shift D = 1 up
✅Answer: A — Amplitude 3, period π, phase shift π/4 right, vertical shift 1 up
Common ACT Math Mistakes to Avoid
❌Mistake: Confusing amplitude with vertical shift
✅Fix: Amplitude is the coefficient in front, vertical shift is the constant added at the end
❌Mistake: Using wrong period formula for tangent (using 2π instead of π)
✅Fix: Remember tangent has period π, sine and cosine have period 2π
❌Mistake: Getting phase shift direction backwards
✅Fix: In y = sin(x - C), shift is C units to the right (positive C means right)
❌Mistake: Mixing up horizontal and vertical transformations
✅Fix: Changes inside parentheses affect x (horizontal), changes outside affect y (vertical)
Practice Question — Try It Yourself
The function y = -2 sin(3x + π) - 4 has which amplitude and period?
A) Amplitude 2, period 2π/3
B) Amplitude -2, period 2π/3
C) Amplitude 2, period 6π
D) Amplitude 4, period 2π/3
E) Amplitude 2, period 3π
Show Answer
Answer: A — Amplitude is |A| = |-2| = 2, period is 2π/3 = 2π/3
Key Takeaways for the ACT
Amplitude is always positive — take absolute value of coefficient
Period of sine/cosine = 2π/B, period of tangent = π/B
Phase shifts: subtract means right, add means left
ACT math graphs of trig functions often test multiple transformations in one equation
With 5 answer choices on the ACT, eliminate options that have wrong amplitude or period first
Related ACT Math Topics
Strengthen your ACT math prep with these related topics:
Trig identities →
Unit circle →