Math / Trigonometry

Question12. How many different sinusoidal functions can be written that have a period of $3 \pi$ , an amplitude of 1 , and a minimum at $(2,3)$ ? [1 mark]

Studdy Solution

STEP 1

1. The sinusoidal function is of the form $y = a \sin(bx + c) + d$ or $y = a \cos(bx + c) + d$ .
2. The period of the function is $3\pi$ .
3. The amplitude of the function is $1$ .
4. The function has a minimum at the point $(2, 3)$ .

STEP 2

1. Determine the general form of the sinusoidal function.
2. Use the given period to find the value of $b$ .
3. Use the given amplitude to determine $a$ .
4. Use the minimum point to find $c$ and $d$ .
5. Count the number of different sinusoidal functions that satisfy these conditions.

STEP 3

Determine the general form of the sinusoidal function. Since the amplitude is 1, we have:
$y = \sin(bx + c) + d \quad \text{or} \quad y = \cos(bx + c) + d$

STEP 4

Use the given period to find $b$ . The period $T$ is given by:
$T = \frac{2\pi}{|b|}$
Given $T = 3\pi$ , solve for $b$ :
$3\pi = \frac{2\pi}{|b|}$ $|b| = \frac{2}{3}$

STEP 5

Use the given amplitude to determine $a$ . Since the amplitude is 1, we have:
$|a| = 1$
Thus, $a = 1$ or $a = -1$ .

STEP 6

Use the minimum point to find $c$ and $d$ . The minimum at $(2, 3)$ implies:
For $y = \sin(bx + c) + d$ :
$\sin(b \cdot 2 + c) = -1 \quad \text{and} \quad d = 4$
For $y = \cos(bx + c) + d$ :
$\cos(b \cdot 2 + c) = -1 \quad \text{and} \quad d = 4$
Determine $c$ for each case:
For $\sin$ , $b \cdot 2 + c = \frac{3\pi}{2} + 2k\pi$ for integer $k$ .
For $\cos$ , $b \cdot 2 + c = \pi + 2k\pi$ for integer $k$ .

STEP 7

Count the number of different sinusoidal functions. Each choice of $a$ and each integer $k$ gives a different function. Since $a = 1$ or $a = -1$ , and $k$ can be any integer, there are infinitely many functions.
The number of different sinusoidal functions is:
$\boxed{\infty}$

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Question12. How many different sinusoidal functions can be written that have a period of $3 \pi$ , an amplitude of 1 , and a minimum at $(2,3)$ ? [1 mark]

Studdy Solution

STEP 1

1. The sinusoidal function is of the form $y = a \sin(bx + c) + d$ or $y = a \cos(bx + c) + d$ .
2. The period of the function is $3\pi$ .
3. The amplitude of the function is $1$ .
4. The function has a minimum at the point $(2, 3)$ .

STEP 2

1. Determine the general form of the sinusoidal function.
2. Use the given period to find the value of $b$ .
3. Use the given amplitude to determine $a$ .
4. Use the minimum point to find $c$ and $d$ .
5. Count the number of different sinusoidal functions that satisfy these conditions.

STEP 3

Determine the general form of the sinusoidal function. Since the amplitude is 1, we have:
$y = \sin(bx + c) + d \quad \text{or} \quad y = \cos(bx + c) + d$

STEP 4

Use the given period to find $b$ . The period $T$ is given by:
$T = \frac{2\pi}{|b|}$
Given $T = 3\pi$ , solve for $b$ :
$3\pi = \frac{2\pi}{|b|}$ $|b| = \frac{2}{3}$

STEP 5

Use the given amplitude to determine $a$ . Since the amplitude is 1, we have:
$|a| = 1$
Thus, $a = 1$ or $a = -1$ .

STEP 6

STEP 7

Count the number of different sinusoidal functions. Each choice of $a$ and each integer $k$ gives a different function. Since $a = 1$ or $a = -1$ , and $k$ can be any integer, there are infinitely many functions.
The number of different sinusoidal functions is:
$\boxed{\infty}$

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Question12. How many different sinusoidal functions can be written that have a period of 3π3 \pi3π, an amplitude of 1 , and a minimum at (2,3)(2,3)(2,3) ? [1 mark]

Studdy Solution

STEP 1

STEP 2

1. Determine the general form of the sinusoidal function.2. Use the given period to find the value of b b b.3. Use the given amplitude to determine a a a.4. Use the minimum point to find c c c and d d d.5. Count the number of different sinusoidal functions that satisfy these conditions.

STEP 3

Determine the general form of the sinusoidal function. Since the amplitude is 1, we have:y=sin⁡(bx+c)+dory=cos⁡(bx+c)+d y = \sin(bx + c) + d \quad \text{or} \quad y = \cos(bx + c) + d y=sin(bx+c)+dory=cos(bx+c)+d

STEP 4

Use the given period to find b b b. The period T T T is given by:T=2π∣b∣ T = \frac{2\pi}{|b|} T=∣b∣2π​Given T=3π T = 3\pi T=3π, solve for b b b:3π=2π∣b∣ 3\pi = \frac{2\pi}{|b|} 3π=∣b∣2π​ ∣b∣=23 |b| = \frac{2}{3} ∣b∣=32​

STEP 5

Use the given amplitude to determine a a a. Since the amplitude is 1, we have:∣a∣=1 |a| = 1 ∣a∣=1Thus, a=1 a = 1 a=1 or a=−1 a = -1 a=−1.

STEP 6

STEP 7

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Question12. How many different sinusoidal functions can be written that have a period of $3 \pi$ , an amplitude of 1 , and a minimum at $(2,3)$ ? [1 mark]

1. Determine the general form of the sinusoidal function.
2. Use the given period to find the value of $b$ .
3. Use the given amplitude to determine $a$ .
4. Use the minimum point to find $c$ and $d$ .
5. Count the number of different sinusoidal functions that satisfy these conditions.

Determine the general form of the sinusoidal function. Since the amplitude is 1, we have:
$y = \sin(bx + c) + d \quad \text{or} \quad y = \cos(bx + c) + d$

Use the given period to find $b$ . The period $T$ is given by:
$T = \frac{2\pi}{|b|}$
Given $T = 3\pi$ , solve for $b$ :
$3\pi = \frac{2\pi}{|b|}$ $|b| = \frac{2}{3}$

Use the given amplitude to determine $a$ . Since the amplitude is 1, we have:
$|a| = 1$
Thus, $a = 1$ or $a = -1$ .