In the standard (x,y) coordinate plane, the graph of a trigonometric function is given by … — ACT

In the standard $(x,y)$ coordinate plane, the graph of a trigonometric function is given by $f(x) = 4 \sin\left(3x - \frac{\pi}{2}\right) + 2$ . What is the horizontal distance between any two consecutive points where the graph intersects its midline $y = 2$ ?

A
$\frac{\pi}{6}$
$\frac{\pi}{3}$ Answer
C
$\frac{\pi}{2}$
D
$\frac{2\pi}{3}$
E
$\frac{4\pi}{3}$

Answer

The horizontal distance between consecutive midline intersections is

\frac{\pi}{3}

The midline of

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

is the horizontal line

y = 2

. Intersections with this line occur when

\sin\left(3x - \frac{\pi}{2}\right) = 0

. Since the sine function equals zero at integer multiples of

\pi

, the difference in the argument between consecutive zero points is

\pi

. Setting

3\Delta x = \pi

gives

\Delta x = \frac{\pi}{3}

, which is half the period of the function.

Step-by-Step Solution

Identify the frequency parameter

B

from the function

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

The frequency coefficient inside the sine expression is

B = 3

The standard form of a transformed sine function is

f(x) = A \sin(Bx - C) + D

Calculate the full period

T

of the function.

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

The standard period

2\pi

of a sine function is scaled horizontally by a factor of

\frac{1}{B}

Determine the horizontal distance between consecutive intersections with the midline

y = 2

\text{Distance} = \frac{T}{2} = \frac{\frac{2\pi}{3}}{2} = \frac{\pi}{3}.

A sinusoidal wave completes one full period over length

T

and intersects its midline twice during each full cycle, making consecutive midline crossings separated by half of the period.

Key Concept

Distance between consecutive midline intersections of a sinusoidal function

Estimated Time:1m 0s

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