In the standard (x,y) coordinate plane, what is the distance along the x-axis between any … — ACT

In the standard $(x,y)$ coordinate plane, what is the distance along the $x$ -axis between any two consecutive maximum points on the graph of the function $f(x) = 5 \cos\left(3x - \frac{\pi}{2}\right) + 4$ ?

A
$\frac{\pi}{3}$
B
$\frac{\pi}{2}$
$\frac{2\pi}{3}$ Answer
D
$2\pi$
E
$6\pi$

Answer

The distance between any two consecutive maximum points is

\frac{2\pi}{3}

The distance along the

x

-axis between consecutive peaks of a cosine wave represents one full period. For the function

f(x) = 5 \cos\left(3x - \frac{\pi}{2}\right) + 4

, the coefficient of

x

is 3. The period formula for cosine is

\frac{2\pi}{|B|}

, which evaluates to

\frac{2\pi}{3}

Step-by-Step Solution

Relate consecutive maximum points to the function's period

The distance along the

x

-axis between consecutive maximum points of a periodic trigonometric function equals one period length,

T

Cosine graphs repeat their peak values once per complete wave cycle.

Identify the coefficient

B

of the variable

x

f(x) = 5 \cos\left(3x - \frac{\pi}{2}\right) + 4

, the coefficient of

x

B = 3

The standard transformation model is

y = A \cos(Bx - C) + D

Calculate the period using

T = \frac{2\pi}{|B|}

T = \frac{2\pi}{3}

Dividing the base cosine period of

2\pi

|B| = 3

gives the period of the transformed function.

Key Concept

The period of a transformed cosine function

y = A \cos(Bx - C) + D

\frac{2\pi}{|B|}

, which measures the horizontal distance between consecutive peak values.

Question

Answer

Step-by-Step Solution

Key Concept

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