Match each transformed trigonometric function on the left with its set of defining graphic… — ACT

Match each transformed trigonometric function on the left with its set of defining graphical properties on the right.

$f(x) = 4\sin(3x) + 2$ Amplitude of $4$ , period of $\frac{2\pi}{3}$ , and a midline at $y = 2$
$g(x) = -2\cos\left(2x - \pi\right)$ Amplitude of $2$ , period of $\pi$ , and a phase shift of $\frac{\pi}{2}$ units to the right
$h(x) = 3\tan\left(\frac{1}{2}x\right) - 1$ Period of $2\pi$ , vertical shift of $1$ unit down, with vertical asymptotes at $x = \pi + 2k\pi$ for any integer $k$
$k(x) = \cos\left(x + \frac{\pi}{3}\right) - 4$ Midline at $y = -4$ , amplitude of $1$ , and a phase shift of $\frac{\pi}{3}$ units to the left

Answer

The correct pairings are:

f(x) = 4\sin(3x) + 2

matches with 'Amplitude of 4, period of 2π/3, and a midline at y = 2';

g(x) = -2\cos(2x - π)

matches with 'Amplitude of 2, period of π, and a phase shift of π/2 units to the right';

h(x) = 3\tan(x/2) - 1

matches with 'Period of 2π, vertical shift of 1 unit down, with vertical asymptotes at x = π + 2kπ'; and

k(x) = \cos(x + π/3) - 4

matches with 'Midline at y = -4, amplitude of 1, and a phase shift of π/3 units to the left'.

Each trigonometric function is matched according to its standard parameter transformation rules:

y = A\sin(B(x - C)) + D

y = A\cos(B(x - C)) + D

, where

|A|

is amplitude, period is

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

for sine/cosine and

\frac{\pi}{|B|}

for tangent,

C

is horizontal phase shift, and

y = D

is the midline.

Step-by-Step Solution

Analyze

f(x) = 4\sin(3x) + 2

using the standard form

y = A\sin(B(x - C)) + D

Amplitude

= |A| = 4

, period

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

, midline

= D \implies y = 2

Direct extraction of parameters for sine graphs.

Factor out

B = 2

from

g(x) = -2\cos(2x - \pi)

g(x) = -2\cos\left(2\left(x - \frac{\pi}{2}\right)\right)

, so amplitude

= |-2| = 2

, period

= \frac{2\pi}{2} = \pi

, phase shift

= \frac{\pi}{2}

to the right.

Factoring

B

is necessary to correctly identify the horizontal phase shift.

Analyze tangent function parameters for

h(x) = 3\tan\left(\frac{1}{2}x\right) - 1

Period

= \frac{\pi}{B} = \frac{\pi}{1/2} = 2\pi

, shifted down

1

unit (

y = -1

). Asymptotes occur when

\frac{1}{2}x = \frac{\pi}{2} + k\pi \implies x = \pi + 2k\pi

Tangent period uses

\frac{\pi}{B}

instead of

\frac{2\pi}{B}

, and asymptotes occur where tangent arguments equal odd multiples of

\frac{\pi}{2}

Analyze transformation parameters for

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

Amplitude

= 1

, midline

= y = -4

, phase shift

= \frac{\pi}{3}

to the left.

Addition inside the function argument

(x + C)

corresponds to a horizontal shift to the left.

Key Concept

Identifying amplitude, period, midline, phase shift, and asymptotes from transformed trigonometric equations

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