The average monthly temperatures of Atlanta, GA are shown below.

Jan 32 degrees
Feb 35 degrees
Mar 44 degrees
Apr 53 degrees
May 63 degrees
June 73 degrees
July 77 degrees
Aug 76 degrees
Sept 69 degrees
Oct 57 degrees
Nov 47 degrees
Dec 37 degrees
Determine the amplitude, period, phase shift, and vertical shift of a sinusoidal function that models the monthly temperatures using x = 1 to represent January.


Write an equation of a sinusoidal function that models the monthly temperatures.


According to your model, what is Atlanta’s average temperature in July? December?

ok, one more. Then you really must provide some work.

You have
min = 32, max = 77
amplitude = (max-min)/2 = 45/2
center line = (max+min)/2 = 109/2
So far, you have
y = 45/2 sinx + 109/2
The period is clearly 12 months, so since sin(kx) has period 2π/k, we have
2π/k = 12
h = π/6
Now we are down to
y = 45/2 sin(π/6 (x-h)) + 109/2
Now we just need the phase shift. Since cosx has a maximum at x=0, and your graph has a maximum at x=7,
y = 45/2 cos(π/6 (x-7)) + 109/2

Now just plug in your month number to see how closely it fits the table.

Well, if I had to model monthly temperatures in Atlanta, I'd do it with a smile! Let's dig in and have some fun with math, shall we?

To find the amplitude, we need to identify the difference between the highest and lowest points of the sinusoidal function. In this case, it goes from 32 to 77, so the amplitude is 77 - 32 = 45.

Now, let's calculate the period. The period is the length it takes for the sinusoidal function to complete one full cycle. Since we're working with months as our x-axis unit, the period would be 12 months.

Next up, we'll determine the phase shift. The phase shift is the horizontal shift of the sinusoidal function compared to the standard sine or cosine function. In this case, since x = 1 represents January, there is no horizontal shift, so the phase shift is 0.

Finally, we'll look at the vertical shift, which refers to the vertical displacement of the function. In this case, it seems that the average temperature is centered around 55 degrees. Therefore, the vertical shift is 55.

So, the equation of the sinusoidal function that models the monthly temperatures in Atlanta would be:

T(x) = 45sin((2π/12)x) + 55

Now, it's time to put on our clown noses and plug in the numbers for July and December!

For July, x would be 7 since it represents the seventh month. Plugging it into our equation, we get:

T(7) = 45sin((2π/12) * 7) + 55

T(7) = 45sin((7π/6)) + 55

Calculating that, we find that Atlanta's average temperature in July according to our model would be approximately 73 degrees.

For December, x would be 12, the twelfth month. Plugging it into our equation, we get:

T(12) = 45sin((2π/12) * 12) + 55

T(12) = 45sin((2π)) + 55

Since the sine function is periodic with a period of 2π, we know that sin(2π) = 0. Therefore:

T(12) = 45 * 0 + 55

T(12) = 55

According to our model, Atlanta's average temperature in December would be approximately 55 degrees.

And there you have it! A bit of math and a bit of humor to model Atlanta's temperatures and predict what July and December might bring. Stay cool and warm, my friend!

To determine the amplitude, period, phase shift, and vertical shift of the sinusoidal function, we can use the given monthly temperatures:

Amplitude: The amplitude of the sinusoidal function is the distance from the midline to the maximum or minimum value. In this case, the maximum temperature is 77 degrees in July, and the minimum is 32 degrees in January. The midline is the average of these values, which is (77 + 32) / 2 = 54.5 degrees. So the amplitude is |77 - 54.5| = 22.5 degrees.

Period: The period of a sinusoidal function is the length of one complete cycle. In this case, the cycle repeats every 12 months. Therefore, the period is 12.

Phase Shift: The phase shift of a sinusoidal function represents any horizontal shift of the graph. Since x = 1 represents January, the sinusoidal function is not shifted horizontally. Therefore, the phase shift is 0.

Vertical Shift: The vertical shift represents any vertical shift of the graph. The midline of this sinusoidal function is 54.5 degrees. Since the midline is the horizontal line that represents the average temperature, the vertical shift is also 54.5 degrees.

Therefore, the equation of the sinusoidal function that models the monthly temperatures is:

f(x) = 22.5 * sin((2π/12) * (x - 1)) + 54.5

To find Atlanta's average temperature in July, we can substitute x = 7 into the equation:

f(7) = 22.5 * sin((2π/12) * (7 - 1)) + 54.5

Calculating this expression, we find:

f(7) ≈ 22.5 * sin((2π/12) * 6) + 54.5
≈ 22.5 * sin(π) + 54.5
≈ 22.5 * 0 + 54.5
≈ 54.5

So Atlanta's average temperature in July is approximately 54.5 degrees.

Similarly, to find Atlanta's average temperature in December, we can substitute x = 12 into the equation:

f(12) = 22.5 * sin((2π/12) * (12 - 1)) + 54.5

Calculating this expression, we find:

f(12) ≈ 22.5 * sin((2π/12) * 11) + 54.5
≈ 22.5 * sin((11π)/6) + 54.5

Using a calculator, we can evaluate sin((11π)/6) as approximately -0.866:

f(12) ≈ 22.5 * (-0.866) + 54.5
≈ -19.47 + 54.5
≈ 35.03

So Atlanta's average temperature in December is approximately 35.03 degrees.

To determine the amplitude, period, phase shift, and vertical shift of a sinusoidal function that models the monthly temperatures, we can analyze the given data.

1. Amplitude: The amplitude of a sinusoidal function represents the vertical distance from the average value to the maximum or minimum value. To find the amplitude, we need to find the range of the given data, which is the difference between the maximum and minimum values. In this case, the maximum value is 77 degrees (in July), and the minimum value is 32 degrees (in January). Therefore, the range is 77 - 32 = 45 degrees. Since the sinusoidal function is centered around the average value, we divide the range by 2 to get the amplitude: 45 / 2 = 22.5 degrees.

2. Period: The period of a sinusoidal function represents the horizontal distance between two consecutive peak or trough points. In this case, we can observe that the pattern repeats every 12 months (from January to December). Therefore, the period is 12.

3. Phase Shift: The phase shift of a sinusoidal function represents the horizontal displacement of the graph. Since x = 1 represents January, which is the starting point of the pattern, there is no phase shift.

4. Vertical Shift: The vertical shift represents the average value or the vertical displacement of the graph. It can be calculated by finding the sum of the maximum and minimum values and dividing it by 2. In this case, (77 + 32) / 2 = 109 / 2 = 54.5 degrees.

Therefore, the sinusoidal function that models the monthly temperatures can be written as:

f(x) = 22.5 * sin((2π/12)(x - 1)) + 54.5

To find Atlanta's average temperature in July, we substitute x = 7 (since July is the 7th month) into the equation:

f(7) = 22.5 * sin((2π/12)(7 - 1)) + 54.5 = 22.5 * sin(2π/3) + 54.5

Using a calculator, we find that sin(2π/3) is approximately 0.866, so:

f(7) ≈ 22.5 * 0.866 + 54.5 ≈ 58.74 degrees

Therefore, according to this model, Atlanta's average temperature in July is approximately 58.74 degrees.

Similarly, to find Atlanta's average temperature in December, we substitute x = 12 into the equation:

f(12) = 22.5 * sin((2π/12)(12 - 1)) + 54.5 = 22.5 * sin(π) + 54.5

Since sin(π) = 0, we get:

f(12) = 0 + 54.5 = 54.5 degrees

Therefore, according to this model, Atlanta's average temperature in December is 54.5 degrees.