Math Lesson 7:
Trigonometric Functions

Graphs of Trigonometric Functions

Trigonometric functions are fundamental in mathematics and have various real-world applications. These functions are periodic, meaning they repeat their values in regular intervals.

The general form of a trigonometric function is:

$y = a \cdot \sin(bx + p) + q$  or  $y = a \cdot \cos(bx + p) + q$

Where:

• a affects the amplitude (height)
• b affects the period (frequency)
• p affects the horizontal shift (phase shift)
• q affects the vertical shift

The Effect of 'a': Change in Amplitude

The amplitude is the distance from the midline to a maximum or minimum point. For $y = a \sin(x)$ or $y = a \cos(x)$:

• If $|a| > 1$, the amplitude increases (vertical stretch).
• If $0 < |a| < 1$, the amplitude decreases (vertical compression).
• If $a < 0$, the graph is reflected over the x-axis.

The Effect of 'q': Vertical Shift

The '$q$' value causes a vertical translation of the entire graph:

• If $q > 0$, the graph shifts up.
• If $q < 0$, the graph shifts down.

The Effect of 'b': Change in Period

The period is the distance required for one full cycle. For $y = \sin(bx)$ or $y = \cos(bx)$:

• The period is calculated as: Period = $\frac{360^\circ}{|b|}$ or $\frac{2\pi}{|b|}$
• If $|b| > 1$, the period decreases (horizontal compression).
• If $0 < |b| < 1$, the period increases (horizontal stretch).

The Effect of 'p': Horizontal Shift

The '$p$' value causes a horizontal shift (phase shift):

• If $p > 0$, the graph shifts left.
• If $p < 0$, the graph shifts right.

AI Interaction