In the standard (x,y) coordinate plane, a sinusoidal function f(x) = A \sin(Bx - C) + D re… — ACT

In the standard $(x,y)$ coordinate plane, a sinusoidal function $f(x) = A \sin(Bx - C) + D$ reaches a maximum value of $9$ at $x = 1$ and its immediate next minimum value of $-1$ at $x = 4$ . What is the period of $f(x)$ ?

Answer: 6

Answer

The period of the sinusoidal function is 6.

For any sinusoidal graph, the horizontal distance between a maximum point and the immediate next minimum point corresponds to one-half of the period. Given that the maximum occurs at

x = 1

and the next minimum occurs at

x = 4

, the half-period is

4 - 1 = 3

. Multiplying this half-period by

2

gives the full period of

6

Step-by-Step Solution

Identify the horizontal distance between the consecutive maximum and minimum points.

The horizontal distance is

4 - 1 = 3

The maximum occurs at

x = 1

and the consecutive minimum occurs at

x = 4

Relate the horizontal distance between consecutive extrema to the period of the function.

Half of the period is equal to

3

In any sinusoidal function, the horizontal distance between a peak (maximum) and the adjacent trough (minimum) represents exactly one-half of a complete cycle.

Calculate the full period of the function.

Period =

2 \times 3 = 6

Multiplying the half-period by 2 yields the full period of the function.

Key Concept

The horizontal distance between consecutive maximum and minimum points of a sinusoidal graph is half of the function's period.

Estimated Time:1m 15s

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