### Student Page 3.5.2 Linear Functions and Fitness Plans

By now, you are proficient at finding equations. In the student page, Linear Functions and Fitness Plans, you will use tables to locate an interval over which two or more of the fitness plans cost the same amount. You will refine your search for an intersection point with graphs and then by solving pairs of equations.

#### 1.

Consider payment plans for three different fitness centers:

Shape Up charges an initiation fee of $39 plus $35 per month per person.

Fitness Fun charges an initiation fee of $75 plus $20 per month per person.

Be Your Best has no initiation fee and charges $52 per month per person.

Complete Table 3.5.2.1 showing the total cost for the first 6 months for each fitness company's payment plan (initiation fees are only charged once).

Amount Paid so far (in dollars) | |||
---|---|---|---|

Month | Shape Up | Fitness Fun | Be Your Best |

1 | 74 | ||

2 | 109 | ||

3 | |||

4 | |||

5 | |||

6 | |||

\(m\) |

#### 2.

For each set of data, graph (month, amount paid so far). Label each graph with the company's name.

#### 3.

##### (a)

Which company charges the most after 6 months of membership? How do you know?

##### (b)

How can you tell from a graph?

##### (c)

How can you tell from the table?

#### 4.

Which company has the steepest graph? Why does that make sense?

#### 5.

What does the \(y\)-intercept represent for each fitness payment plan?

#### 6.

Find an equation that fits each data set. Include the equation in the last row of Table 3.5.2.1.

#### 7.

You have $500 to spend on a fitness plan this year. Show your work.

##### (a)

Which fitness plans can you afford?

##### (b)

For how many months can you afford each fitness plan?

#### 8.

In the summer, you are very active outside. You only want to join a fitness center for 7, 8, or 9 months of the year. The fitness centers offer the same amenities with one exception, Be Your Best also has a pool.

##### (a)

Determine the cost for all three fitness plans for these numbers of months. Fill in Table 3.5.2.2

Amount Paid so far ($) | |||
---|---|---|---|

Month | Shape Up | Fitness Fun | Be Your Best |

7 | |||

8 | |||

9 |

##### (b)

Which fitness plan would you choose and for how many months? Explain.

#### 9.

##### (a)

What variable are you solving for in Student Page Exercise 3.5.2.7? How can you tell?

##### (b)

What variable are you solving for in Student Page Exercise 3.5.2.8? How can you tell?

#### 10.

##### (a)

Solve each equation you found in Student Page Exercise 3.5.2.6 for the variable that represents months.

Shape Up:

Fitness Fun:

Be Your Best:

##### (b)

What do these equations allow you to do directly that the equations in Student Page Exercise 3.5.2.6 require more work to do?

#### 11.

Consider the following pairs of fitness plans. For what number of months will you pay the same amount for a membership in either plan?

##### (a)

Shape Up and Fitness Fun:

##### (b)

Fitness Fun and Be Your Best:

##### (c)

Be Your Best and Shape Up: