Analyzing Sinusoidal Graphs: Amplitude, Period, and Solutions Explained

Understanding Amplitude, Midline, and Period from Graph Points

Amplitude is the vertical distance from the midline to a maximum or minimum point. It can be calculated as half the difference between the maximum and minimum values: [ \text{Amplitude} = \frac{\text{Maximum} - \text{Minimum}}{2} ]
Midline (D) is the vertical midpoint between the maximum and minimum points: [ D = \frac{\text{Maximum} + \text{Minimum}}{2} ]
Period (P) is the horizontal length of one full cycle of the wave. It equals twice the horizontal distance between a maximum and the next minimum point: [ P = 2 \times (\text{Horizontal distance between max and min}) ]
The parameter ( B ) in the sinusoidal function ( f(x) = A \sin(Bx) + D ) or ( f(x) = A \cos(Bx) + D ) relates to the period by: [ B = \frac{360^\circ}{P} ]

Example: Modeling a Cosine Function

Given a curve with:

Maximum point at ( (1, 3.5) )
Minimum point at ( (2, 0.5) )
( a < 0 ) (amplitude negative)

Calculate:

Amplitude ( |a| = \frac{3.5 - 0.5}{2} = 1.5 )
Midline ( c = \frac{3.5 + 0.5}{2} = 2 )
Period ( P = 2 \times (2 - 1) = 2 )
( B = \frac{360}{2} = 180 )

Function form: [ f(t) = -1.5 \cos(180t) + 2 ]

Solving for Specific Values

Number of Solutions for ( f(t) = 2 )

The horizontal line ( y = 2 ) intersects the graph 6 times within the domain ( t \in [1, 5] ).

Number of Solutions for ( f(t) = 0.5 )

The line ( y = 0.5 ) intersects the graph 3 times.

Solving ( f(t) = 3 ) for ( t \in [0, 2] )

Set up the equation: [ 1.5 \cos(180t) + 2 = 3 ]
Using a graphing calculator, find intersections at:
- ( t_1 = 0.732 )
- ( t_2 = 1.27 )

Solving Inequality ( f(t) \leq 1.5 ) for ( t \in [0, 2] )

Inequality: [ 1.5 \cos(180t) + 2 \leq 1.5 ]
Graphing reveals solutions in intervals:
- ( t \in [0, 0.392] )
- ( t \in [1.61, 2] )

Using a Graphing Calculator

Enter the function ( f(t) = 1.5 \cos(180t) + 2 ) in the graphing tool.
Adjust window settings:
- ( x_{min} = 0 ), ( x_{max} = 2 )
- ( y_{min} = -1 ), ( y_{max} = 5 )
- Scale appropriately for clear viewing.
Use intersection features to find exact solution points.
Change display settings to auto for precise decimal values.

Summary

This tutorial covers how to extract key parameters from sinusoidal graphs and solve related equations and inequalities. It emphasizes the relationship between maximum/minimum points and amplitude, midline, and period, and demonstrates practical graphing techniques to find solutions within specified domains.

For further understanding of trigonometric functions, you may find these resources helpful: