Neil has been running a tutoring business since 2005. He charges a monthly fee for weekly tutoring sessions and a phone help line. Each year, he has increased his fee by the same amount. The table shows what Neil charged each customer for two given years of his business:
Year Annual Tutoring Fee
2005 $1200
2008 $1350
A. What is the rate of change and initial value for Neil's business? How do you know?
B. Write an equation in slope-intercept form to represent the fees that Neil charges each year. (10 points)
A. To find the rate of change, we need to calculate the difference in fees between the two years and divide it by the number of years that have passed.
Rate of change = (Fee in 2008 - Fee in 2005) / (2008 - 2005)
Rate of change = ($1350 - $1200) / (2008 - 2005)
Rate of change = $150 / 3 = $50
The rate of change is $50 per year.
To find the initial value, we need to determine what Neil charged in the first year of his business, which is represented by the fee in 2005.
The initial value is $1200.
B. We can use the slope-intercept form of a linear equation, y = mx + b, to represent the fees that Neil charges each year.
y = annual tutoring fee
x = number of years
From our calculations above, we know that the rate of change (m) is $50 and the initial value (b) is $1200.
Therefore, the equation is:
y = 50x + 1200