For the trigonometric equations on the left, which description on the right correctly matc… — ACT

For the trigonometric equations on the left, which description on the right correctly matches the graphical features of each equation?

$y = -3\sin\left(2x - \frac{\pi}{3}\right) + 1$ A function with a period of $\pi$ , a range of $[-2, 4]$ , and a positive $y$ -intercept.
$y = 2\cos\left(\frac{1}{2}x + \frac{\pi}{4}\right) - 1$ A function with a period of $4\pi$ , a range of $[-3, 1]$ , and a phase shift of $\frac{\pi}{2}$ units to the left.
$y = -\tan\left(3x - \frac{\pi}{2}\right) + 2$ A function with a period of $\frac{\pi}{3}$ that is undefined at the $y$ -axis.
$y = 3\sin\left(\frac{2}{3}x + \pi\right) - 2$ A function with a period of $3\pi$ , a range of $[-5, 1]$ , and a $y$ -intercept at $(0, -2)$ .

Answer

The correct matches are: the equation

y = -3\sin\left(2x - \frac{\pi}{3}\right) + 1

matches the description with a period of

\pi

, a range of

[-2, 4]

, and a positive

y

-intercept; the equation

y = 2\cos\left(\frac{1}{2}x + \frac{\pi}{4}\right) - 1

matches the description with a period of

4\pi

, a range of

[-3, 1]

, and a phase shift of

\frac{\pi}{2}

units to the left; the equation

y = -\tan\left(3x - \frac{\pi}{2}\right) + 2

matches the description with a period of

\frac{\pi}{3}

that is undefined at the

y

-axis; and the equation

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

matches the description with a period of

3\pi

, a range of

[-5, 1]

, and a

y

-intercept of

(0, -2)

Each equation is correctly paired with the unique graphical properties determined by calculating its amplitude, period, range, phase shift, and

y

-intercept. The first equation features a period of

\pi

, a range of

[-2, 4]

, and a positive

y

-intercept. The second equation has a period of

4\pi

, a range of

[-3, 1]

, and a phase shift of

\frac{\pi}{2}

units left. The third equation has a period of

\frac{\pi}{3}

and is undefined at the

y

-axis since

x=0

creates a vertical asymptote. The fourth equation has a period of

3\pi

, a range of

[-5, 1]

, and a

y

-intercept of

(0, -2)

Step-by-Step Solution

Determine the period, range, and y-intercept for the function

y = -3\sin\left(2x - \frac{\pi}{3}\right) + 1

The period is

\pi

, range is

[-2, 4]

, and

y

-intercept is

1 + \frac{3\sqrt{3}}{2} > 0

The period is

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

, the range is

[1-3, 1+3] = [-2, 4]

, and evaluating at

x=0

yields

y = -3\sin(-\pi/3) + 1 > 0

Determine the period, range, and phase shift for the function

y = 2\cos\left(\frac{1}{2}x + \frac{\pi}{4}\right) - 1

The period is

4\pi

, range is

[-3, 1]

, and phase shift is

\frac{\pi}{2}

units left.

The period is

\frac{2\pi}{1/2} = 4\pi

, the range is

[-1-2, -1+2] = [-3, 1]

, and factoring the argument yields

\frac{1}{2}(x + \frac{\pi}{2})

, giving a shift of

\frac{\pi}{2}

units left.

Determine the period and domain restriction for the function

y = -\tan\left(3x - \frac{\pi}{2}\right) + 2

The period is

\frac{\pi}{3}

and the function is undefined at

x = 0

(the

y

-axis).

For tangent, the period is

\frac{\pi}{3}

. At

x=0

, the argument is

-\frac{\pi}{2}

, where tangent is undefined, meaning the function has a vertical asymptote at the

y

-axis.

Determine the period, range, and y-intercept for the function

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

The period is

3\pi

, range is

[-5, 1]

, and

y

-intercept is

(0, -2)

The period is

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

, the range is

[-2-3, -2+3] = [-5, 1]

, and at

x=0

y = 3\sin(\pi) - 2 = -2

Key Concept

Identifying graphs of trigonometric functions from their equations by determining amplitude, period, phase shift, midline, range, and asymptotes.

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