Question

the table provided shows the amounts (in billions $) spent on admission to movie theaters in the united states for the years 1997 through 2003.
find the quadratic model of the data set provided. round to the nearest thousandth.
model: ( f ( x ) = ) type your answer... ( x ^ { 2 } + ) type your answer... ( x + ) type your answer...

Explanation:

Step1: Assign \(x\) - values

Let \(x = 0\) represent the year 1997. Then:

• For 1997: \(x = 0\)
• For 1998: \(x = 1\)
• For 1999: \(x = 2\)
• For 2000: \(x = 3\)
• For 2001: \(x = 4\)
• For 2002: \(x = 5\)
• For 2003: \(x = 6\)

The corresponding \(y\) - values (amounts in billions) are:

• When \(x = 0\), \(y=6.3\)
• When \(x = 1\), \(y = 6.9\)
• When \(x = 2\), \(y=7.9\)
• When \(x = 3\), \(y = 8.6\)
• When \(x = 4\), \(y=9.0\)
• When \(x = 5\), \(y = 9.6\)
• When \(x = 6\), \(y=9.9\)

Step2: Use the quadratic regression formula

The quadratic regression formula is \(y=ax^{2}+bx + c\). Using a calculator or software for quadratic regression (e.g., in Excel: Data - Data Analysis - Regression (if the Analysis ToolPak is installed), or using a graphing calculator's quadratic regression function):
Input the \(x\) and \(y\) values.
The quadratic regression formula gives \(a\approx0.121\), \(b\approx0.307\), \(c\approx6.3\)

Answer:

\(f(x)=0.121x^{2}+0.307x + 6.3\)