Barry owns a music store, He recorded the sales of a newly launched album as part of his biweekly inventory. The table shows the number of weeks, x, since the launch of the album and the number of albums sold during that time period, f(x).

x 0 2 4 6 8

f(x) 0 78 152 222 288

Use the data in the table to create the standard form of the function that models this situation.

f(x)___ = x2 +_ x + _____

[tex]f(x) = ax^{2} + bx + c[/tex], where we have to determine the values of unknowns such as a, b and c. We need three equations to determine three unknowns such as a, b, and c.

Using orders pairs from data table we can determine the values of a, b, and c.

As the table consists of four ordered pairs : (0, 0), (2, 78), (4, 152), (6, 222) and (8, 288).

Lets take the ordered pair (0, 0), and put the value in [tex]f(x) = ax^{2} + bx + c[/tex].

So,

[tex]f(x) = ax^{2} + bx + c[/tex]

[tex]0 = a(0)^{2} + b(0) + c[/tex]

[tex]c = 0[/tex]

Lets take the ordered pair (4, 152), and put the value in [tex]f(x) = ax^{2} + bx + c[/tex].

So,

[tex]f(x) = ax^{2} + bx + c[/tex]

[tex]152 = a(4)^{2} + b(4) + 0[/tex]

[tex]152 = 16a+ 4b + 0[/tex]

[tex]152 = 16a+ 4b[/tex]

Dividing the above equation by 4

[tex]38 = 4a+ b[/tex] → Equation (A)

Lets take the ordered pair (6, 222), and put the value in [tex]f(x) = ax^{2} + bx + c[/tex].

So,

[tex]f(x) = ax^{2} + bx + c[/tex]

[tex]222 = a(6)^{2} + b(6) + 0[/tex]

[tex]222 = 36a+ 6b + 0[/tex]

[tex]222 = 36a+ 6b[/tex]

Dividing the above equation by 6

[tex]37 = 6a+ b[/tex] → Equation (B)

Subtract Equation (B) i.e. [tex]38 = 4a+ b[/tex] from

Equation (A) i.e. [tex]38 = 4a+ b[/tex]

[tex]1 = -2a[/tex]

[tex]a = -0.5[/tex]

Entering the value of [tex]a = -0.5[/tex] in [tex]37 = 6a+ b[/tex]

[tex]37 = 6(-0.5)+ b[/tex]

[tex]37 = -3 + b[/tex]

[tex]b = 40[/tex]

Hence, the function will be: [tex]f(x) = -0.5x^{2} + 40x [/tex]

Keywords: quadratic equation, function

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