Developing Explicit Formulas Mountain Bike Rental Developing Explicit Formulas

Rental Cost: Mountain Bike Rental While on vacation, you want to rent a mountain bike. Rental Cost: There is a $10 rental fee, plus a $5 charge for each hour used.

1. What would be the cost of renting the bike for 12 hours? 2. If you have a $100 budget for the bike rental, how many hours can you use it?

Rental Cost: Mountain Bike Rental There is a $10 rental fee, plus a $5 charge for each hour used.

Let’s take a closer look.

Record your data in a table for 0, 1, 2, … , 12 rental hours.

Record your data in a table for 0, 1, 2, … , 12 rental hours. Graph the data.

Record your data in a table for 0, 1, 2, … , 12 rental hours. Graph the data. Describe the process in words.

Record your data in a table for 0, 1, 2, … , 12 rental hours. Graph the data. Describe the process in words.

c0 = ? cn = ? Mountain Bike Rental Define a RECURSIVE FORMULA for this situation, which will predict the cost of the rental, for a given a number of hours. c0 = ? cn = ? Define your variables. Does c0 have meaning by itself? What is its meaning in the context of this situation?

cn = ? An EXPLICIT FORMULA gives a direct relationship between two quantities. cn = ?

The EXPLICIT FORUMLA for this situation can be found by examining the recursive pattern. cn = ?

cn = 10 + 5n An EXPLICIT FORMULA 10 = the initial value gives a direct relationship between two quantities. cn = 10 + 5n 10 = the initial value 5 = the rate of change

1. What would be the cost of renting the bike for 12 hours? 2. If you have a $100 budget for the bike rental, how many hours can you use it?

1. What would be the cost of renting the bike for 12 hours? cn = 10 + 5n c12 = ?

2. If you have a $100 budget for the bike rental, how many hours can you use it? cn = 10 + 5n cn = 100 n = ?

2. If you have a $100 budget for the bike rental, how many hours can you use it? cn = 10 + 5n cn = 100 n = ? 100 = 10 + 5n

Let’s examine the graph.

What type of graph is modeled by this data?

linear data What is the y-intercept of this line? What is the line’s slope (rate of change)? Compare these values to your recursive formula and your explicit formula.

c0 = 10 c0 = the initial value linear data The y-intercept of the line (0, 10) linear data The y-intercept of the line corresponds to the initial value of the recursive pattern. c0 = the initial value

3 hr $15 1 hr $5 linear data For an arithmetic sequence, the rate of change is the SLOPE of the line. SLOPE: 5 dollars 1 hour

y = a + bx u0 = a un = a + bn un = un–1 + b Linear Equation a = initial value b = slope (rate of change) Recursive Formula u0 = a Explicit Formula un = a + bn un = un–1 + b

y = a + bx y = 10 + 5x y = mx + b y = 5x + 10 Linear Equation Important Note: The different uses of the variable b other textbooks: Linear Equation y = mx + b y = 5x + 10

Applying this to Sequences Given the arithmetic sequence: 52, 73, 94, 115, … 1. Find u0 2. Represent the sequence three ways: 3. Find u29 4. Find n if un = 1186

Applying this to Sequences 52, 73, 94, 115, … 1. Find u0 n un 1 2 3 4 5 6 2. Find u29 3. Find n if un = 1186

Applying this to Sequences 52, 73, 94, 115, … 1. Find u0 n un 1 52 2 73 3 94 4 115 5 6 2. Find u29 3. Find n if un = 1186

Applying this to Sequences 52, 73, 94, 115, … 1. Find u0 n un 1 52 2 73 3 94 4 115 5 136 6 157 2. Find u29 3. Find n if un = 1186

Applying this to Sequences 52, 73, 94, 115, … 1. Find u0 n un 31 1 52 2 73 3 94 4 115 5 136 6 157 2. Find u29 3. Find n if un = 1186

Practice Given the arithmetic sequence: 34, 53, 72, 91, … 1. Find u0 2. Represent the sequence three ways: 3. Find u40 4. Find n if un = 661

Practice Given the arithmetic sequence: 205, 173, 141, 109, … 1. Find u0 2. Represent the sequence three ways: 3. Find u25 4. Find n if un = –275