Math / Algebra

QuestionOver the past several years, the owner of a boutique on Aspen Avenue has observed a pattern in the amount of revenue for the store. The revenue reaches a maximum of about $\$ 56000$ in January and a minimum of about $\$ 25000$ in July. Suppose the months are numbered 1 through 12 , and write a function of the form $f(x)=A \sin (B[x-C])+D$ that models the boutique's revenue during the year, where $x$ corresponds to the month. $f(x)=$ $\square$

Studdy Solution

STEP 1

What is this asking? We need to find a sine function that describes how much money a store makes each month, hitting a high of $\$56000$ in January ( $x = 1$ ) and a low of $\$25000$ in July ( $x = 7$ ). Watch out! Remember that sine waves go up and down, so we need to make sure our function starts at the right spot and has the right height and width!

STEP 2

1. Find the midline.
2. Find the amplitude.
3. Find the period and $B$ .
4. Find the phase shift.
5. Build the function.

STEP 3

The midline is the average of the highest and lowest revenue values.
Think of it like the middle of the sine wave!
It's where the wave would be if it were flat.

STEP 4

$D = \frac{56000 + 25000}{2} = \frac{81000}{2} = 40500$ So, our midline is $D = 40500$ .

STEP 5

The amplitude is how far the highest and lowest points are from the midline.
It's like measuring how tall the wave is from its middle!

STEP 6

$A = \frac{56000 - 25000}{2} = \frac{31000}{2} = 15500$ Our amplitude is $A = 15500$ .

STEP 7

The period of a sine wave is how long it takes for it to complete one full cycle.
Since revenue goes from max to min and back to max in 12 months, our period is 12.

STEP 8

The period is related to $B$ by the formula $\text{Period} = \frac{2\pi}{B}$ .
We know the period is 12, so we can solve for $B$ : $12 = \frac{2\pi}{B}$ $B = \frac{2\pi}{12} = \frac{\pi}{6}$ So, $B = \frac{\pi}{6}$ .

STEP 9

The phase shift is how much the sine wave is shifted horizontally.
Since the maximum revenue occurs in January ( $x = 1$ ), which is when a regular sine wave is at its maximum, we can use a sine function without a phase shift.
This means $C = 1$ .

STEP 10

Now, we plug all our values into the formula $f(x) = A\sin(B[x - C]) + D$ : $f(x) = 15500\sin\left(\frac{\pi}{6}(x - 1)\right) + 40500$

STEP 11

$f(x) = 15500\sin\left(\frac{\pi}{6}(x - 1)\right) + 40500$

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Studdy Solution

STEP 1

STEP 2

1. Find the midline.
2. Find the amplitude.
3. Find the period and $B$ .
4. Find the phase shift.
5. Build the function.

STEP 3

The midline is the average of the highest and lowest revenue values.
Think of it like the middle of the sine wave!
It's where the wave would be if it were flat.

STEP 4

$D = \frac{56000 + 25000}{2} = \frac{81000}{2} = 40500$ So, our midline is $D = 40500$ .

STEP 5

The amplitude is how far the highest and lowest points are from the midline.
It's like measuring how tall the wave is from its middle!

STEP 6

$A = \frac{56000 - 25000}{2} = \frac{31000}{2} = 15500$ Our amplitude is $A = 15500$ .

STEP 7

The period of a sine wave is how long it takes for it to complete one full cycle.
Since revenue goes from max to min and back to max in 12 months, our period is 12.

STEP 8

The period is related to $B$ by the formula $\text{Period} = \frac{2\pi}{B}$ .
We know the period is 12, so we can solve for $B$ : $12 = \frac{2\pi}{B}$ $B = \frac{2\pi}{12} = \frac{\pi}{6}$ So, $B = \frac{\pi}{6}$ .

STEP 9

The phase shift is how much the sine wave is shifted horizontally.
Since the maximum revenue occurs in January ( $x = 1$ ), which is when a regular sine wave is at its maximum, we can use a sine function without a phase shift.
This means $C = 1$ .

STEP 10

Now, we plug all our values into the formula $f(x) = A\sin(B[x - C]) + D$ : $f(x) = 15500\sin\left(\frac{\pi}{6}(x - 1)\right) + 40500$

STEP 11

$f(x) = 15500\sin\left(\frac{\pi}{6}(x - 1)\right) + 40500$

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Studdy Solution

STEP 1

STEP 2

1. Find the midline.2. Find the amplitude.3. Find the period and BBB.4. Find the phase shift.5. Build the function.

STEP 3

The midline is the average of the **highest** and **lowest** revenue values.Think of it like the middle of the sine wave!It's where the wave would be if it were flat.

STEP 4

D=56000+250002=810002=40500 D = \frac{56000 + 25000}{2} = \frac{81000}{2} = 40500 D=256000+25000​=281000​=40500 So, our **midline** is D=40500D = 40500D=40500.

STEP 5

The amplitude is how far the highest and lowest points are from the midline.It's like measuring how tall the wave is from its middle!

STEP 6

A=56000−250002=310002=15500 A = \frac{56000 - 25000}{2} = \frac{31000}{2} = 15500 A=256000−25000​=231000​=15500 Our **amplitude** is A=15500A = 15500A=15500.

STEP 7

The period of a sine wave is how long it takes for it to complete one full cycle.Since revenue goes from max to min and back to max in 12 months, our period is **12**.

STEP 8

The period is related to BBB by the formula Period=2πB \text{Period} = \frac{2\pi}{B} Period=B2π​.We know the period is **12**, so we can solve for BBB: 12=2πB 12 = \frac{2\pi}{B} 12=B2π​ B=2π12=π6 B = \frac{2\pi}{12} = \frac{\pi}{6} B=122π​=6π​So, B=π6B = \frac{\pi}{6}B=6π​.

STEP 9

The phase shift is how much the sine wave is shifted horizontally.Since the maximum revenue occurs in January (x=1x = 1x=1), which is when a regular sine wave is at its maximum, we can use a sine function without a phase shift.This means C=1C = 1C=1.

STEP 10

Now, we plug all our values into the formula f(x)=Asin⁡(B[x−C])+Df(x) = A\sin(B[x - C]) + Df(x)=Asin(B[x−C])+D: f(x)=15500sin⁡(π6(x−1))+40500 f(x) = 15500\sin\left(\frac{\pi}{6}(x - 1)\right) + 40500 f(x)=15500sin(6π​(x−1))+40500

STEP 11

f(x)=15500sin⁡(π6(x−1))+40500 f(x) = 15500\sin\left(\frac{\pi}{6}(x - 1)\right) + 40500 f(x)=15500sin(6π​(x−1))+40500

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1. Find the midline.
2. Find the amplitude.
3. Find the period and $B$ .
4. Find the phase shift.
5. Build the function.

The midline is the average of the highest and lowest revenue values.
Think of it like the middle of the sine wave!
It's where the wave would be if it were flat.

$D = \frac{56000 + 25000}{2} = \frac{81000}{2} = 40500$ So, our midline is $D = 40500$ .

The amplitude is how far the highest and lowest points are from the midline.
It's like measuring how tall the wave is from its middle!

$A = \frac{56000 - 25000}{2} = \frac{31000}{2} = 15500$ Our amplitude is $A = 15500$ .

The period of a sine wave is how long it takes for it to complete one full cycle.
Since revenue goes from max to min and back to max in 12 months, our period is 12.

The period is related to $B$ by the formula $\text{Period} = \frac{2\pi}{B}$ .
We know the period is 12, so we can solve for $B$ : $12 = \frac{2\pi}{B}$ $B = \frac{2\pi}{12} = \frac{\pi}{6}$ So, $B = \frac{\pi}{6}$ .

The phase shift is how much the sine wave is shifted horizontally.
Since the maximum revenue occurs in January ( $x = 1$ ), which is when a regular sine wave is at its maximum, we can use a sine function without a phase shift.
This means $C = 1$ .

Now, we plug all our values into the formula $f(x) = A\sin(B[x - C]) + D$ : $f(x) = 15500\sin\left(\frac{\pi}{6}(x - 1)\right) + 40500$

$f(x) = 15500\sin\left(\frac{\pi}{6}(x - 1)\right) + 40500$