Weekend Bike Repair: Comparing Functions

Compare rates, starting values, and outputs across different representations.

Compare Labor Charges

Amira and Leo run a weekend bike-repair business. Compare their proposed labor charges. Each model is linear.

For both proposals, x is the number of labor hours and y is the total charge in dollars. Amira's proposal is shown in the table. Leo's proposal is y = 7x + 3.

Amira's Labor Proposal
Labor hours	Total charge ($)
0	9
2	19
5	34

Which statement correctly compares the hourly rates?

A. Leo's rate is $2 per hour greater.B. Amira's rate is $6 per hour greater.C. Leo's rate is $6 per hour greater.D. Amira's rate is $2 per hour greater.

How much greater is Amira's starting charge than Leo's? $

A repair needs 2 hours of labor. Which proposal costs less, and by how much?

A. Amira's proposal costs $2 less.B. Leo's proposal costs $4 less.C. Leo's proposal costs $2 less.D. Amira's proposal costs $6 less.

For how many labor hours do the two proposals charge the same total? hours

Compare Cleaning Earnings

The students also test two bike-cleaning services. Compare their net earnings after subtracting supply costs. Both models are linear.

For this section, x is the number of bikes cleaned and y is net earnings in dollars. Negative earnings mean the service has not yet recovered its supply cost. The graph shows Amira's model; the table shows Leo's model.

Leo's Cleaning Model
Bikes cleaned	Net earnings ($)
0	−6
2	0
4	6

How much greater is Leo's increase in net earnings per bike than Amira's? $ per bike

Which statement correctly compares their starting net earnings?

A. Amira starts $2 higher than Leo.B. Amira starts $4 higher than Leo.C. Leo starts $6 higher than Amira.D. Leo starts $2 higher than Amira.

What do the two models have in common?

A. They have the same earnings increase per bike.B. They reach zero net earnings after cleaning 2 bikes.C. They have the same earnings before cleaning bikes.D. They earn the same amount after cleaning 4 bikes.

Each student cleans 3 bikes using their own service.

How much greater are Leo's net earnings than Amira's? $

Make Business Decisions

Each question presents a new pair of models for the bike-repair business. Compare the models using the same input.

Amira proposes a tire-service charge of y = 4x + 8, where x is the number of tires serviced. Leo proposes a $5 starting charge plus $4 for each tire. Which comparison is correct?

A. Their charges grow equally, but Leo starts $3 higher.B. Their charges grow equally, but Amira starts $3 higher.C. Amira's charge grows faster and starts $3 higher.D. Leo's charge grows faster and starts $3 lower.

10.

For tool savings, Amira starts with $18 and adds $6 each weekend. Leo's linear savings model is shown below. Which comparison is correct?

Leo's Tool Savings
Weekends elapsed	Savings ($)
1	23
3	33
5	43

A. They start with the same amount, but Leo saves faster.B. Amira starts with more, but both save at the same rate.C. They start with the same amount, but Amira saves faster.D. Leo starts with more, but Amira saves faster.

11.

Two suppliers offer replacement cables. The table shows the total cost from the local supplier. The online supplier charges y = 2x + 7 dollars for x cables, including shipping. Both models are linear.

Local Cable Supplier
Cables	Total cost ($)
2	6
4	12
6	18

For an order of 10 cables, how much less does the online supplier charge? $

12.

The graph shows Amira's number of unfinished repairs after x hours. Leo starts with 7 unfinished repairs and completes 1 repair per hour. Which statement compares the models over the first 3 hours?

A. Amira starts with more repairs and finishes them at a faster rate.B. Leo starts with fewer repairs and finishes them at a faster rate.C. Leo starts with more repairs and finishes them at a faster rate.D. Amira starts with fewer repairs and finishes them at a faster rate.

13.

Amira plans to offer a safety check for $6 per bike with no starting fee. Leo proposes the linear pricing model in the table. Which statement correctly compares the proposals?

Leo's Safety-Check Proposal
Bikes	Total charge ($)
1	10
3	20
5	30

A. Amira charges less for every positive number of bikes.B. They charge equally for 5 bikes, and Amira charges less for 6 bikes.C. Leo charges less for every positive number of bikes.D. They charge equally for 5 bikes, and Leo charges less for 6 bikes.

14.

For a new repair package, Amira proposes y = 8x + 2 dollars for x repairs. Leo proposes a $14 starting charge plus $5 per repair. Leo says, “Amira's rate per repair is higher, so her total charge is always higher.” Explain why this claim is incorrect. Use totals for two different numbers of repairs as evidence.