In the standard (x,y) coordinate plane, a cosine function of the form y = a \cos(b(x - c))… — ACT

In the standard $(x,y)$ coordinate plane, a cosine function of the form $y = a \cos(b(x - c)) + d$ , where $a > 0$ and $b > 0$ , has a local maximum at $(2, 9)$ and the very next local minimum at $(6, 1)$ . What is the value of $b$ ?

A
$\frac{\pi}{8}$
$\frac{\pi}{4}$ Answer
C
$\frac{\pi}{2}$
D
$\pi$
E
$2\pi$

Answer

\frac{\pi}{4}

The distance along the

x

-axis from a maximum to the consecutive minimum is half of one full period of the cosine wave. Here, that distance is

6 - 2 = 4

, which means the full period is

2 \times 4 = 8

. Using the relationship

\text{Period} = \frac{2\pi}{b}

, we solve for

b

to get

b = \frac{2\pi}{8} = \frac{\pi}{4}

Step-by-Step Solution

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

The horizontal distance is

6 - 2 = 4

units.

The

x

-coordinates of the maximum and minimum points are

2

and

6

, respectively.

Calculate the period of the cosine function.

\text{Period} = 2 \times 4 = 8

units.

The horizontal distance between a peak and the immediately following trough of a cosine wave represents exactly half of one full period.

Solve for the coefficient

b

using the period formula

b = \frac{2\pi}{\text{Period}}

b = \frac{2\pi}{8} = \frac{\pi}{4}

For a trigonometric function of the form

y = a \cos(b(x - c)) + d

, the relationship between the period and

b

\text{Period} = \frac{2\pi}{b}

Key Concept

Determining Period and Frequency Coefficient of a Cosine Function

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