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Question

A sports store chain estimates weekly sales of soccer balls (in thousands of units) n months after an International Soccer Tournament as a parabolic function. Sales totalled 7,000 units 1 month after the tournament and peaked 2 months after the tournament at 17,000 units.

a. Determine the function that describes soccer ball sales following the International Soccer Tournament.

Answer 1

A parabolic function is of the form:

y = ax² + bx + c

in this case it is more convenient to write

s = an² + bn + c

s⁽¹⁾ = 2an + b

s is in thousands of units. Therefore

s(1) = a(1)² + b(1) + c = 7

s(2) = b(2)² + b(2) + c = 17

We also know s(2) is where the function peaked, which means s⁽¹⁾(2) = 0 or the first derivative of the function is 0 when n = 2

s⁽¹⁾ (2) = 2a(2) + b = 4a + b = 0

You now have 3 equations with 3 variables and you can do the algebra to solve for a, b, and c

Answer 2

Let y be the number of units sold after x months. Since it peaked after two months, that's the vertex of the parabola, thus we have y = a(x - 2)² + 17000. We now have to solve for the constant a. Since there were 7000 units after 1 month we can solve 7000 = a(1 - 2)²+ 17000 which implies that a = -10000.

Our function is thus y = -10000(x - 2)² + 17000

Answer 3

Can set up vertex form for parabola equation

x-axis is months; y-axis is no. of balls (000's)

y = a(x - h)² + k where h and k are the vertex coordinates. As sales peak at 17,000 2 months after the tournament h = 2 and k = 17

Need to find a (constant). Use the other given info of 7000 sales 1 month after. (1, 7)

7 = a(1 - 2)² + 17 → 7 = a + 17 → a = -10

So equation is y = -10(x - 2)² + 17 (all values in 000's)

Check it on the Desmos Graphing Calculator website.