Question

1. a research firm determined that the equation ( y = 120 + 6x ) can be used to model the total amount of spending by tourists in florida, ( y ), in billions of dollars, for the years 1992 - 2001 where ( x ) is the number of years since 1992.

a. determine the key features for the context. show any necessary work below the table.

domain

range

as ( x )-values increase, ( y )-values...

slope

( x )-intercept

( y )-intercept

Explanation:

Step1: Identify the equation type

The equation is \( y = 120 + 6x \), which is a linear equation in slope - intercept form \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.

Step2: Find the domain

The variable \( x \) represents the number of years since 1992, and the time period is 1992 - 2001. So \( x \) ranges from 0 (1992) to 9 (2001, since \( 2001 - 1992=9 \)). So the domain is \( \{x|0\leq x\leq9, x\in\mathbb{Z}\} \) (or in interval notation \([0, 9]\) if we consider non - integer years, but since it's years since 1992, usually we take integer values from 0 to 9).

Step3: Find the range

To find the range, we substitute the minimum and maximum values of \( x \) into the equation. When \( x = 0 \), \( y=120 + 6(0)=120 \). When \( x = 9 \), \( y=120+6(9)=120 + 54 = 174 \). Since the slope \( m = 6>0 \), the function is increasing, so the range is \( \{y|120\leq y\leq174, y\in\mathbb{R}\} \) (or \([120, 174]\) in interval notation).

Step4: Analyze as \( x \) increases

Since the slope \( m = 6>0 \), as \( x \) values increase, \( y \) values increase.

Step5: Find the slope

For the linear equation \( y=mx + b \), comparing with \( y = 6x+120 \), the slope \( m = 6 \).

Step6: Find the x - intercept

To find the x - intercept, set \( y = 0 \) and solve for \( x \).
\( 0=120 + 6x \)
\( 6x=- 120 \)
\( x=- 20 \). But in the context of the problem (years since 1992), \( x=-20 \) is not in our domain (since \( x\geq0 \) for the period 1992 - 2001), so in the context, there is no x - intercept within the domain.

Step7: Find the y - intercept

The y - intercept is the value of \( y \) when \( x = 0 \). From the equation \( y = 120+6x \), when \( x = 0 \), \( y = 120 \). So the y - intercept is 120.

Answer:

• Domain: \( 0\leq x\leq9 \) (where \( x \) is the number of years since 1992)
• Range: \( 120\leq y\leq174 \) (where \( y \) is in billions of dollars)
• As \( x \) values increase: \( y \) values increase (since slope \( m = 6>0 \))
• Slope: \( 6 \)
• x - intercept: In the context of 1992 - 2001, there is no x - intercept (since \( x=-20 \) is outside the domain). Mathematically, \( x=-20 \)
• y - intercept: \( 120 \)