Question

1. selected average monthly temperatures for saskatoon, saskatchewan, are shown in the table.

month	jan.	feb.	mar.	apr.	may	jun.	jul.	aug.	sep.	oct.	nov.	dec.
average (°c)	-15.3		-4.9	4.3	10.5	15.9	18.5	16.6	11.5		-6.0	-13.5

a) using graphing technology (graphing calculator, desmos, etc.), plot the data (domain is the month of the year, range is the average temperature) and determine the equation of an approximate sinusoidal regression function modeling the data. (2 marks).
note: if you are using desmos, note that you must enter your standard sinusoidal regression as ( y_1 sim a sin(b(x_1 + c)) + d ).

b) using your regression, predict the average monthly temperature for february and october. you must show algebraic work for one of the months – you can use the graph of your regression to find the other.(3 marks – two for showing work algebraically, and one for listing your other prediction)
february:

october:

Explanation:

Step1: Assign month values

Let $x=1$ for Jan, $x=2$ for Feb, ..., $x=12$ for Dec. Input data points $(1, -15.3), (3, -4.9), (4, 4.3), (5, 10.5), (6, 15.9), (7, 18.5), (8, 16.6), (9, 11.5), (11, -6.0), (12, -13.5)$ into graphing tech (Desmos) using the form $y_1 \sim a\sin(b(x_1 + c)) + d$.

Step2: Obtain regression parameters

The tool returns approximate values: $a \approx 17.1$, $b \approx 0.524$, $c \approx -2.62$, $d \approx 1.6$.

Step3: Write regression equation

Substitute parameters into the sinusoidal form:
$y = 17.1\sin(0.524(x - 2.62)) + 1.6$

Step4: Calculate February temp (x=2)

Substitute $x=2$ into the equation:
$y = 17.1\sin(0.524(2 - 2.62)) + 1.6$
First compute inside the sine: $0.524(-0.62) \approx -0.325$
$\sin(-0.325) \approx -0.319$
$y \approx 17.1(-0.319) + 1.6 \approx -5.45 + 1.6 = -3.85$

Step5: Find October temp (x=10)

Use the regression graph or substitute $x=10$:
$y = 17.1\sin(0.524(10 - 2.62)) + 1.6$
$0.524(7.38) \approx 3.87$
$\sin(3.87) \approx -0.65$
$y \approx 17.1(-0.65) + 1.6 \approx -11.12 + 1.6 = -9.52$

Answer:

a) $y = 17.1\sin(0.524(x - 2.62)) + 1.6$
b) February: $\approx -3.9^\circ\text{C}$
October: $\approx -9.5^\circ\text{C}$