Math Lesson 7:
 Trigonometric Functions

Graphs of Trigonometric Functions

Trigonometric functions are fundamental in mathematics and have various real-world applications. In this section, we'll explore the graphs of sine, cosine, and tangent functions.

The general form of a trigonometric function is: y = a * sin(bx + p) + q or y = a * cos(bx + p) + q

Where:

  • a affects the amplitude
  • b affects the period
  • p affects the horizontal shift
  • q affects the vertical shift

Exercise: Identifying Basic Trig Graphs

Match the following functions to their correct graph:

  1. y = sin(x)
  2. y = cos(x)
  3. y = tan(x)

The Effect of 'a' on the Shape of the Graph: Change in Amplitude

The amplitude of a trigonometric function is the distance from the midline to a maximum or minimum point. The value of 'a' in the function affects this amplitude.

For y = a * sin(x) or y = a * cos(x):

  • If |a| > 1, the amplitude increases
  • If 0 < |a| < 1, the amplitude decreases
  • If a < 0, the graph is reflected over the x-axis

Exercise: Amplitude Changes

Describe how the amplitude of y = sin(x) changes for each of these functions:

  1. y = 2sin(x)
  2. y = 0.5sin(x)
  3. y = -sin(x)

The Effect of 'q' on the Shape of the Graph: Vertical Shift

The 'q' value in a trigonometric function causes a vertical shift of the entire graph.

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

Exercise: Vertical Shifts

Describe the vertical shift for each of these functions:

  1. y = sin(x) + 2
  2. y = cos(x) - 1
  3. y = tan(x) + 0.5

The Effect of 'b' on the Shape of the Graph: Change in Period

The period of a trigonometric function is the distance required for the function to complete one full cycle. The 'b' value affects this period.

For y = sin(bx) or y = cos(bx):

  • The period is given by (2π / |b|)
  • If |b| > 1, the period decreases (graph compresses horizontally)
  • If 0 < |b| < 1, the period increases (graph stretches horizontally)

Exercise: Period Changes

Calculate the period for each of these functions:

  1. y = sin(2x)
  2. y = cos(0.5x)
  3. y = tan(πx)

The Effect of 'p' on the Shape of the Graph: Horizontal Shift

The 'p' value in a trigonometric function causes a horizontal shift of the entire graph.

  • For y = sin(x + p) or y = cos(x + p):
  • If p > 0, the graph shifts left
  • If p < 0, the graph shifts right

Exercise: Horizontal Shifts

Describe the horizontal shift for each of these functions:

  1. y = sin(x + π/2)
  2. y = cos(x - π)
  3. y = tan(x + π/4)