IB Mathematics AA Trigonometric Graph Study Notes

IB Mathematics AA Trigonometric Graph Study Notes

IB Mathematics AA Trigonometric Graph Study Notes

IB Mathematics AA Trigonometric Graph Study Notes Offer a clear explanation of Trigonometric Graph , including various formula, rules, exam style questions as example to explain the topics. Worked Out  examples and common problem types provided here will be sufficient to cover for topic Trigonometric Graph.

Circular Functions sin x, cos x, tan x

Circular Functions sin x, cos x, tan x

 

Sine function: \( y = \sin x \)

  •  Amplitude: 1
  •  Period: \( 2\pi \)
  •  Range: \( [-1, 1] \)

Cosine function: \( y = \cos x \)

  • Amplitude: 1 
  • Period: \( 2\pi \) 
  • Range: \( [-1, 1] \)

Tangent function: \( y = \tan x \)

  • No amplitude (unbounded) 
  • Period: \( \pi \) 
  • Range: \( (-\infty, \infty) \) 
  • Vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \)

Amplitude: The maximum vertical distance from the midline of the wave.

For \( y = A \sin x \) or \( y = A \cos x \), amplitude = \( |A| \).

Periodicity: The length of one full cycle of the function.

For \( y = \sin(Bx) \) or \( y = \cos(Bx) \), period = \( \frac{2\pi}{|B|} \). For \( y = \tan(Bx) \), period = \( \frac{\pi}{|B|} \).

Example 

Sketch the graph of \( y = 2 \sin x \) and state its amplitude and period.

▶️Answer/Explanation
  • Amplitude = \( |2| = 2 \)
  • Period = \( 2\pi \)
  • The graph oscillates between -2 and 2 with a period of \( 2\pi \).

Example 

Sketch the graph of \( y = \tan x \) for \( -\pi \le x \le \pi \).

▶️Answer/Explanation
  • Period = \( \pi \)
  • Vertical asymptotes at \( x = -\frac{\pi}{2}, \frac{\pi}{2} \)
  • The graph passes through the origin and repeats every \( \pi \).

Composite Trigonometric Functions

Composite Trigonometric Functions

A function of the form: \[ f(x) = a \sin(b(x + c)) + d \] represents a transformed sine wave.

  • Amplitude: \( |a| \)
  • Period: \( \frac{2\pi}{|b|} \) (for sine or cosine)
  • Phase shift: \( -c \)
  • Vertical translation: \( d \)

Example 

Analyze \( f(x) = \tan(x – \frac{\pi}{4}) \)

▶️Answer/Explanation
  • Standard form: \( \tan(b(x + c)) + d \)
  • Here: \( b = 1 \), \( c = -\frac{\pi}{4} \), \( d = 0 \)
  • Period: \( \pi \)
  • Phase shift: \( \frac{\pi}{4} \)
  • Vertical shift: 0
  • The graph of \( \tan x \) shifts right by \( \frac{\pi}{4} \).

   

Example 

Analyze \( f(x) = 2 \cos(3(x – 4)) + 1 \)

▶️Answer/Explanation
  • Amplitude: \( |2| = 2 \)
  • Period: \( \frac{2\pi}{3} \)
  • Phase shift: \( 4 \)
  • Vertical shift: \( +1 \)
  • The cosine graph is stretched vertically, compressed horizontally, shifted right by 4 units, and moved up by 1 unit.

Transformations of a General Function

Transformations of a General Function

For any function \( y = f(x) \), transformations can be written as:

Vertical stretch/compression: \( y = a f(x) \)

Scale factor \( |a| \) in the y-direction.

Horizontal stretch/compression: \( y = f(bx) \)

Scale factor \( \frac{1}{|b|} \) in the x-direction.

Vertical translation: \( y = f(x) + d \)

Shifts graph up/down by \( d \).

Horizontal translation: \( y = f(x + c) \)

Shifts graph left by \( c \) (if \( c > 0 \)) or right (if \( c < 0 \)).

Reflections:

\( y = -f(x) \) reflects over x-axis;
\( y = f(-x) \) reflects over y-axis.

Example

Describe the transformation from \( y = \sin x \) to \( y = 3 \sin 2x \).

▶️Answer/Explanation
  • Vertical stretch:
    The factor 3 stretches the graph in the y-direction by a scale factor of 3.
  • Horizontal compression:
    The factor 2 compresses the graph in the x-direction by a scale factor of \( \frac{1}{2} \).
  • Final result:
    The sine wave has amplitude 3 and a period of \( \pi \) instead of \( 2\pi \).

Trigonometric Models in Real-Life Contexts

Trigonometric Models in Real-Life Contexts

Trigonometric functions are widely used to model periodic phenomena in real life, such as:

  • Tides – height of the water level varies sinusoidally with time.
  • Ferris wheel motion – height of a seat changes over time following a sinusoidal curve.
  • Simple Harmonic Motion (SHM) – displacement of a particle oscillating back and forth can be modeled by \( f(t) = a \sin(bt + c) + d \).

Example: 

A particle moves in SHM. Its displacement from equilibrium at time \( t \) seconds is given by:

\( f(t) = 5 \sin\left( 2t \right) \)

▶️Answer/Explanation
  • Amplitude: \( 5 \) – maximum displacement from equilibrium is 5 units.
  • Angular frequency: \( 2 \) rad/s – the particle completes cycles faster than standard \( \sin t \).
  • Period: \( T = \frac{2\pi}{2} = \pi \) seconds – time for one complete oscillation.
  • Graph: A sine wave with amplitude 5 and period \( \pi \).

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