Question

watch the video and then solve the problem given below.
click here to watch the video.
the table shows the sales (in billions of dollars) of hepatitis c treatments for a certain region from 2012 to 2016. complete parts (a) through (c) below.
a. what is the quadratic function that best models these sales as a function of the number of years after 2010? let x represent the number of years after 2010 and y represent the number of billions of dollars in sales of treatments.
y = (□)x² + (□)x + (□)
(type an integer or decimal rounded to three decimal places as needed.)

Explanation:

Step1: Set up the system of equations

Let the quadratic function be \(y = ax^{2}+bx + c\).
For \(x = 2\) (year 2012), \(y=4.3\), so \(4a + 2b + c=4.3\).
For \(x = 3\) (year 2013), \(y = 4.5\), so \(9a+3b + c=4.5\).
For \(x = 4\) (year 2014), \(y = 6.3\), so \(16a+4b + c=6.3\).

Step2: Subtract equations to eliminate \(c\)

Subtract the first equation from the second: \((9a + 3b + c)-(4a + 2b + c)=4.5 - 4.3\), which simplifies to \(5a + b=0.2\).
Subtract the second equation from the third: \((16a+4b + c)-(9a + 3b + c)=6.3 - 8.2\), which simplifies to \(7a + b=2.1\).

Step3: Solve the new system of equations

Subtract the equation \(5a + b=0.2\) from \(7a + b=2.1\): \((7a + b)-(5a + b)=2.1 - 0.2\), so \(2a=1.9\), then \(a = 0.95\).
Substitute \(a = 0.95\) into \(5a + b=0.2\): \(5\times0.95 + b=0.2\), \(4.75 + b=0.2\), \(b=- 4.55\).
Substitute \(a = 0.95\) and \(b=-4.55\) into \(4a + 2b + c=4.3\): \(4\times0.95+2\times(-4.55)+c=4.3\), \(3.8-9.1 + c=4.3\), \(c = 9.6\).

Answer:

\(y = 0.95x^{2}-4.55x + 9.6\)