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

Hi! Basically, the problem is asking us to find the values of a, b, and c in the equation [tex]f(x)=a x^{2} +bx+c[/tex]. Since we have three unknowns, we just need three equations. We can find these equations by using the data in the table.

First let's plug x = 0 and f(x) = 0.
[tex]0=0+0+c[/tex]
[tex]c=0[/tex]

Now that we know c, it's time to pick two more pairs. Let's plug-in (2,78) and (4,152)

[tex]78=4a+2b[/tex]
[tex]152=16a+4b[/tex]

Before proceeding with the process of eliminating one variable, let us first reduce both equations to their lowest terms. We divide the first equation by 2 and we divide the second one by 4.

[tex]39=2a+b[/tex]
[tex]38=4a+b[/tex]

Next, we subtract equation 2 from equation 1.

[tex]1=-2a[/tex]
[tex]a=-0.5[/tex]

Finally, we substitute the value of a to equation 2 to get the value of b.

[tex]38=4(-0.5)+b[/tex]
[tex]b=40[/tex]

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