To express the relationship between the value of the baseball player's rookie card \( y \) in dollars and the number of years \( x \) the player has been in retirement, we can model this situation with a linear equation.
The value of the card starts at $9.97 when the player retires in 2000 (this is when \( x = 0 \)). The value increases by $1.19 each year.
The general form of the linear equation is:
\[ y = mx + b \]
where:
- \( m \) is the slope (the rate of increase),
- \( x \) is the number of years since retirement,
- \( b \) is the y-intercept (the initial value of the card).
Here:
- The slope \( m = 1.19 \) (the increase in value each year),
- The y-intercept \( b = 9.97 \) (the value of the card when the player retires).
Putting this all together, the equation is:
\[ y = 1.19x + 9.97 \]
Part 2: Is the relationship between \( x \) and \( y \) proportional?
A relationship is proportional if it can be expressed in the form \( y = kx \), where \( k \) is a constant, and the graph of this relationship passes through the origin (0,0). In this case, our equation includes a y-intercept of \( 9.97 \), which means it does not pass through the origin. Therefore, the relationship is not proportional.
Part 3: Finding the value of the card in 2008
To find the value of the card in 2008, we need to determine the number of years \( x \) since retirement in 2000.
By 2008, \( x \) is:
\[ x = 2008 - 2000 = 8 \]
Now, we can substitute \( x = 8 \) into our equation:
\[ y = 1.19(8) + 9.97 \]
Calculating this, we get:
\[ y = 1.19 \times 8 + 9.97 = 9.52 + 9.97 = 19.49 \]
Thus, the value of the card in 2008 was $19.49.
1 answer